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Bernoulli CDF Tails, Limit Continuity Points and Finite-Prefix Limits

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batch-20261014-advance-20261031-r03-convergence-in-distribution-foundations · 2026-10-14 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

1. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=2, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=1, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(1, 31/60, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (1, 31/60, -1, 0, 0, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

2. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=3, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=2, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(2, 13/25, 1/2, 12/25, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (2, 13/25, 1/2, 12/25, 1/2, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

3. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=4, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=3, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(3, 73/140, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (3, 73/140, 2, 1, 1, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

4. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=5, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=4, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(4, 47/90, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (4, 47/90, -1, 0, 0, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

5. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=6, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=5, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(5, 23/44, 1/2, 21/44, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (5, 23/44, 1/2, 21/44, 1/2, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

6. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=7, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=6, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(6, 131/260, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (6, 131/260, 2, 1, 1, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

7. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=8, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=7, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(7, 38/75, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (7, 38/75, -1, 0, 0, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

8. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=9, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=8, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(8, 173/340, 1/2, 167/340, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (8, 173/340, 1/2, 167/340, 1/2, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

9. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=10, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=9, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(9, 97/190, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (9, 97/190, 2, 1, 1, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

10. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=11, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=10, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(10, 43/84, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (10, 43/84, -1, 0, 0, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

11. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=12, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=11, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(11, 231/460, 1/2, 229/460, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (11, 231/460, 1/2, 229/460, 1/2, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

12. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=13, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=12, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(12, 63/125, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (12, 63/125, 2, 1, 1, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

13. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=14, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=13, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(13, 91/180, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (13, 91/180, -1, 0, 0, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

14. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=15, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=14, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(14, 147/290, 1/2, 143/290, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (14, 147/290, 1/2, 143/290, 1/2, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

15. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=16, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=15, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(15, 63/124, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (15, 63/124, 2, 1, 1, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

16. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=17, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=16, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(16, 331/660, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (16, 331/660, -1, 0, 0, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

17. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=18, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=17, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(17, 88/175, 1/2, 87/175, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (17, 88/175, 1/2, 87/175, 1/2, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

18. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=19, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=18, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(18, 373/740, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (18, 373/740, 2, 1, 1, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

19. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=20, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=19, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(19, 197/390, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (19, 197/390, -1, 0, 0, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

20. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=21, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=20, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(20, 83/164, 1/2, 81/164, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (20, 83/164, 1/2, 81/164, 1/2, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

21. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=22, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=21, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(21, 431/860, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (21, 431/860, 2, 1, 1, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

22. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=23, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=22, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(22, 113/225, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (22, 113/225, -1, 0, 0, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

23. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=24, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=23, threshold=1/2. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(23, 473/940, 1/2, 467/940, 1/2, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (23, 473/940, 1/2, 467/940, 1/2, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

24. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=25, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=24, threshold=2/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(24, 247/490, 2, 1, 1, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (24, 247/490, 2, 1, 1, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

25. Compare Bernoulli CDF Values at an Explicit Threshold

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=26, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=25, threshold=-1/1. Give (n,p_n,threshold,F_n,F,limit_CDF_continuity_point?).

View answer and explanation

(25, 103/204, -1, 0, 0, Yes)

Compute p_n exactly, then evaluate both complete CDFs with the same threshold conventions. The final flag describes continuity of the limit CDF, not whether these two finite-n values happen to be equal. The exact result, retaining the requested order, is (25, 103/204, -1, 0, 0, Yes).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-bernoulli-cdf · candidate

26. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=2, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=1, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(1, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (1, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

27. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=3, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=2, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(2, 1/2, 1/50, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (2, 1/2, 1/50, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

28. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=4, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=3, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(3, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (3, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

29. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=5, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=4, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(4, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (4, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

30. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=6, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=5, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(5, 1/2, 1/44, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (5, 1/2, 1/44, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

31. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=7, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=6, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(6, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (6, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

32. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=8, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=7, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(7, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (7, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

33. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=9, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=8, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(8, 1/2, 3/340, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (8, 1/2, 3/340, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

34. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=10, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=9, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(9, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (9, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

35. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=11, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=10, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(10, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (10, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

36. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=12, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=11, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(11, 1/2, 1/460, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (11, 1/2, 1/460, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

37. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=13, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=12, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(12, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (12, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

38. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=14, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=13, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(13, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (13, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

39. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=15, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=14, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(14, 1/2, 1/145, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (14, 1/2, 1/145, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

40. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=16, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=15, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(15, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (15, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

41. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=17, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=16, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(16, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (16, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

42. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=18, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=17, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(17, 1/2, 1/350, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (17, 1/2, 1/350, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

43. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=19, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=18, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(18, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (18, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

44. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=20, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=19, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(19, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (19, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

45. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=21, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=20, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(20, 1/2, 1/164, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (20, 1/2, 1/164, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

46. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=22, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=21, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(21, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (21, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

47. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=23, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=22, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(22, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (22, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

48. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=24, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=23, threshold=1/2. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(23, 1/2, 3/940, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (23, 1/2, 3/940, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

49. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=25, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=24, threshold=2/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(24, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (24, 2, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

50. A Symbolic Parameter Tail Controls Every Required CDF Gap

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=26, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. n=25, threshold=-1/1. Give (n,threshold,absolute_gap,limit_CDF_continuity_point?,gap_tends_to_zero_at_every_limit_CDF_continuity_point).

View answer and explanation

(25, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point)

Take the exact absolute CDF difference. On [0,1) it is |c|/(n+shift); outside that interval it is zero. The explicit all-n tail, not a finite table, supplies convergence at every required continuity point. A jump-point flag does not assert failure of convergence there; the definition simply does not require it. The exact result, retaining the requested order, is (25, -1, 0, Yes, gap_tends_to_zero_at_every_limit_CDF_continuity_point).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-pointwise-gap · candidate

51. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=1/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-12/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(1/30, -12/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/30, -12/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

52. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=2/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-11/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(1/15, -11/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/15, -11/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

53. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=3/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-10/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(1/10, -10/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/10, -10/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

54. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=4/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-9/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(2/15, -9/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (2/15, -9/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

55. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=5/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-8/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(1/6, -8/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/6, -8/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

56. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=6/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-7/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(1/5, -1, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/5, -1, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

57. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=7/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-6/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(7/30, -6/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (7/30, -6/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

58. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=8/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-5/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(4/15, -5/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (4/15, -5/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

59. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=9/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-4/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(3/10, -4/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (3/10, -4/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

60. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=10/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-3/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(1/3, -3/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/3, -3/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

61. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=11/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-2/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(11/30, -2/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (11/30, -2/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

62. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=12/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=-1/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(2/5, -1/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (2/5, -1/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

63. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=13/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=0/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(13/30, 0, No, limit_CDF_jump_no_requirement_from_definition)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (13/30, 0, No, limit_CDF_jump_no_requirement_from_definition).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

64. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=14/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=1/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(7/15, 1/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (7/15, 1/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

65. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=15/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=2/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(1/2, 2/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (1/2, 2/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

66. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=16/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=3/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(8/15, 3/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (8/15, 3/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

67. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=17/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=4/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(17/30, 4/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (17/30, 4/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

68. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=18/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=5/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(3/5, 5/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (3/5, 5/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

69. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=19/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=6/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(19/30, 6/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (19/30, 6/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

70. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=20/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=7/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(2/3, 1, No, limit_CDF_jump_no_requirement_from_definition)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (2/3, 1, No, limit_CDF_jump_no_requirement_from_definition).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

71. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=21/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=8/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(7/10, 8/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (7/10, 8/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

72. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=22/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=9/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(11/15, 9/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (11/15, 9/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

73. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=23/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=10/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(23/30, 10/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (23/30, 10/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

74. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=24/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=11/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(4/5, 11/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (4/5, 11/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

75. Distinguish Required Continuity Points from Limit-CDF Jumps

Use the explicit Bernoulli symbolic family with interior limit p=25/30, 0<p<1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. threshold=12/7. Give (p,threshold,continuity_point?,definition_obligation). The obligation is CDF_convergence_required_here or limit_CDF_jump_no_requirement_from_definition.

View answer and explanation

(5/6, 12/7, Yes, CDF_convergence_required_here)

For the registered interior Bernoulli limit, compare the threshold exactly with 0 and 1. Only the continuity points carry the defining requirement. Degenerate p=0 or p=1 has a different jump set and is not extrapolated from this projection. The exact result, retaining the requested order, is (5/6, 12/7, Yes, CDF_convergence_required_here).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-continuity-obligation · candidate

76. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=2, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=1. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/20, 2, 1, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 2, 1, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

77. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=3, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=2. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/10, 3, 2, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 3, 2, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

78. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=4, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=3. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 3/20, 4, 3, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 4, 3, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

79. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=5, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=4. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/5, 5, 4, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 5, 4, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

80. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=6, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=5. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/4, 6, 5, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 6, 5, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

81. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=7, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=6. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/20, 7, 6, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 7, 6, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

82. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=8, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=7. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/10, 8, 7, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 8, 7, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

83. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=9, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=8. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 3/20, 9, 8, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 9, 8, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

84. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=10, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=9. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/5, 10, 9, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 10, 9, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

85. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=11, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=10. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/4, 11, 10, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 11, 10, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

86. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=12, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=11. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/20, 12, 11, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 12, 11, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

87. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=13, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=12. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/10, 13, 12, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 13, 12, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

88. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=14, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=13. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 3/20, 14, 13, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 14, 13, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

89. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=15, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=14. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/5, 15, 14, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 15, 14, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

90. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=16, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=15. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/4, 16, 15, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 16, 15, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

91. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=17, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=16. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/20, 17, 16, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 17, 16, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

92. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=18, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=17. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/10, 18, 17, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 18, 17, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

93. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=19, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=18. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 3/20, 19, 18, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 19, 18, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

94. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=20, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=19. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/5, 20, 19, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 20, 19, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

95. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=21, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=20. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/4, 21, 20, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 21, 20, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

96. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=1/20, positive shift=22, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=21. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/20, 22, 21, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/20, 22, 21, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

97. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=2/20, positive shift=23, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=22. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/10, 23, 22, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/10, 23, 22, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

98. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=3/20, positive shift=24, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=23. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 3/20, 24, 23, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 3/20, 24, 23, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

99. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=4/20, positive shift=25, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=24. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/5, 25, 24, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/5, 25, 24, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

100. A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Use the explicit Bernoulli symbolic family with interior limit p=1/2, 0<p<1, signed c=5/20, positive shift=26, and p_n=p+c/(n+shift) for every n≥1. The whole parameter tail must stay in [0,1]. The Bernoulli CDF is 0 below 0, 1−p on [0,1), and 1 at or above 1. With this interior limit its only jumps are at 0 and 1; the definition requires CDF convergence only at its continuity points. A chosen coupling is not required, and convergence in probability, almost surely or in Lp is not inferred. Keep exact reduced rationals, omitting /1 for integers. Checked prefix_length=25. Give (p,c,shift,prefix_length,sufficient?,finite_prefix_not_convergence_certificate).

View answer and explanation

(1/2, 1/4, 26, 25, No, finite_prefix_not_convergence_certificate)

The No sufficiency flag concerns a finite-prefix inspection, not the validity of the explicitly declared symbolic family. Its whole tail can supply a separate universal certificate; a long finite prefix alone cannot. The exact result, retaining the requested order, is (1/2, 1/4, 26, 25, No, finite_prefix_not_convergence_certificate).

Reviewed step-by-step derivation

Convergence in distribution · qf-mod31-convdist-finite-prefix · candidate

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