Question 15797: A Symbolic Parameter Tail Controls Every Required CDF Gap

Original question and givens

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).

• G001 — 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).

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, 23)

Bernoulli first legal tail denominator = 24

Bernoulli first tail signed correction := div(1/10, 24)

Bernoulli first tail signed correction = 1/240

Bernoulli first legal parameter := sum(1/2, 1/240)

Bernoulli first legal parameter = 121/240

declared all-n family p_n=1/2+(1/10)/(n+23);n>=1;first parameter=121/240;limit=1/2

for every n>=1,lambda=(1+23)/(n+23) in(0,1];p_n=(1-lambda)*1/2+lambda*(121/240) in[0,1]

Bernoulli selected tail denominator := sum(22, 23)

Bernoulli selected tail denominator = 45

Bernoulli selected signed correction := div(1/10, 45)

Bernoulli selected signed correction = 1/450

Bernoulli selected exact parameter := sum(1/2, 1/450)

Bernoulli selected exact parameter = 113/225

selected distribution:Bernoulli parameter=113/225;same exact threshold=-1;CDF branch=t<0

selected distribution exact CDF value=0

limiting distribution:Bernoulli parameter=1/2;same exact threshold=-1;CDF branch=t<0

limiting distribution exact CDF value=0

Bernoulli selected signed CDF difference := sub(0, 0)

Bernoulli selected signed CDF difference = 0

selected signed CDF difference=0;same threshold=-1

exact absolute gap=0;limit CDF continuity point=Yes

declared all-n parameters p_n=1/2+(1/10)/(n+23);same complete Bernoulli CDF formula

for every real t,gap=abs(1/10)/(n+23) on0<=t<1,and0 outside;for every epsilon>0 choose integer N>=1 with N+23>abs(1/10)/epsilon;then all n>=N give gap<epsilon

Reverse checks

• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S012 · premise_and_result_audit — Recompute Bernoulli selected signed CDF difference with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• 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 selected exact parameter with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Bernoulli selected signed correction with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Bernoulli selected tail 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

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

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.