Question 15753: A Fixed Prefix Does Not Supply the All-n CDF Limit Obligation

Original question and givens

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).

• G001 — 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).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-convergence-in-distribution

Step-by-step derivation

interior Bernoulli limit mass at0 := sub(1, 1/2)

interior Bernoulli limit mass at0 = 1/2

original limit parameter=[1, 2];reduced p=1/2;mass at0=1/2;mass at1=1/2

both endpoint masses positive;only limit-CDF jumps at0 and1;registered interior projection

Bernoulli first legal tail denominator := sum(1, 4)

Bernoulli first legal tail denominator = 5

Bernoulli first tail signed correction := div(3/20, 5)

Bernoulli first tail signed correction = 3/100

Bernoulli first legal parameter := sum(1/2, 3/100)

Bernoulli first legal parameter = 53/100

declared all-n family p_n=1/2+(3/20)/(n+4);n>=1;first parameter=53/100;limit=1/2

for every n>=1,lambda=(1+4)/(n+4) in(0,1];p_n=(1-lambda)*1/2+lambda*(53/100) in[0,1]

Bernoulli last inspected prefix denominator := sum(3, 4)

Bernoulli last inspected prefix denominator = 7

Bernoulli last inspected prefix correction := div(3/20, 7)

Bernoulli last inspected prefix correction = 3/140

Bernoulli last inspected prefix parameter := sum(1/2, 3/140)

Bernoulli last inspected prefix parameter = 73/140

inspected indices1 through3;first legal parameter=53/100;last inspected parameter=73/140

finite-prefix sufficiency=No;finite_prefix_not_convergence_certificate;separate declared whole-tail argument remains available

Reverse checks

• S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S009 · premise_and_result_audit — Recompute Bernoulli last inspected prefix parameter with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Bernoulli last inspected prefix correction with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Bernoulli last inspected prefix denominator with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S005 · premise_and_result_audit — Recompute Bernoulli first legal parameter with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Bernoulli first tail signed correction with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Bernoulli first legal tail denominator with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Recompute interior Bernoulli limit mass at0 with exact rational arithmetic, not rounded decimals.

Result

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

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.