A Bernoulli parameter-tail CDF certificate

Goal and scope

Prove distributional convergence at continuity points of the limiting CDF.

No common coupling is required; no stronger mode is inferred.

Givens and symbols

• work

• G1 — For n≥1, X_n has Bernoulli(p_n) law with p_n=1/3+1/(n+2); X has Bernoulli(1/3) law. Only laws, not a coupling, are declared.

• p_n — Bernoulli parameter
• F_n,F — respective CDFs

Rules and sources

• mko_definition · cited — A common probability space or chosen coupling is not required. A finite prefix does not prove convergence, and the definition imposes no convergence requirement at discontinuities of the limiting CDF. Convergence in distribution does not generally imply convergence in probability, almost surely or in Lp. · mko-convergence-in-distribution
• work · prerequisite — The Bernoulli CDF formula; distributional convergence tests each continuity point of the limit; floor(u)+1>u.

Main derivation

n\ge1

1/3<p_n\le2/3<1

t<0

F_n(t)=F(t)=0

0\le t<1

F_n(t)=1-p_n=2/3-1/(n+2),\quad F(t)=2/3

t\ge1

F_n(t)=F(t)=1

F\text{ is Bernoulli}(1/3)\text{ CDF}

\operatorname{Cont}(F)=\mathbb R\setminus\{0,1\}

0<t<1

|F_n(t)-F(t)|=1/(n+2)

\eta>0,\quad N=\lfloor1/\eta\rfloor+1

n\ge N\implies1/(n+2)<\eta

F_n(t)\to F(t)\quad(t\in\operatorname{Cont}(F))

X_n\Rightarrow X

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — No common space was specified, so this is not stated as convergence in probability.
• S07 · premise_audit · G1, S06, work — Eta is arbitrary positive and N is selected before the tail conclusion.
• S06 · premise_audit · G1, S05, work — The bound covers every t in the interval, not a few fixed points.
• S05 · premise_audit · G1, S04, work — The definition imposes no convergence requirement at those jumps.
• S04 · premise_audit · G1, S03, work — The point 1 belongs to the right branch, although it is a jump point of the limiting CDF.
• S03 · premise_audit · G1, S02, work — Do not use p_n as the CDF on this interval.
• S02 · premise_audit · G1, S01, work — This covers every t<0.
• S01 · premise_audit · G1, work — The maximum occurs at n=1, rather than being estimated from samples.

Error controls and negative controls

Legal parameters

Prove p_n≤2/3 for every n≥1.

Mode boundary

Law convergence does not supply same-space differences.

• All variables share one coin here. → reject — Only laws, not a coupling, were specified.
• Jump points must be checked for the definition. → reject — Only continuity points of the limiting CDF are required.
• A finite set of close CDFs proves convergence. → reject — The proof uses a universal tail for arbitrary eta.

Result and limitations

Bernoulli(1/3+1/(n+2)) converges in distribution to Bernoulli(1/3).

Verification and source binding

2 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 57b8fd580f0c495a9a33f194a0df094ac8136b1ac350e6e98cc85f9a09eff859
• original source SHA-256: 3f7025055c3497aa6b82b00d019258a96a7e7d544c185df0da1bf15436bbdbd0
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)