A Chebyshev tail for fair Bernoulli averages

Goal and scope

Derive the WLLN from a whole-sequence IID premise.

Finite-variance weak law; the conclusion is convergence in probability.

Givens and symbols

• work

• G1 — On a supplied probability space, (X_i)_{i≥1} is an infinite IID sequence: every finite distinct-index joint law factors into Bernoulli(1/2) marginals. This is a whole-process premise, not inferred from finite data.

• bar X_n — n-term sample mean
• N — epsilon-delta tail index

Rules and sources

• mko_definition · cited — Use a declared whole-sequence IID premise, common finite mean mu and variance sigma_squared. Independence gives Var(sample_mean_n)=sigma_squared/n; an exact Chebyshev epsilon-delta tail then gives convergence in probability. Finite simulation is insufficient. · mko-weak-law-large-numbers
• work · prerequisite — Assume Chebyshev, linearity of expectation and variance expansion with covariances; independent finite-second-moment variables have zero covariance. Convergence in probability means that every epsilon>0 and delta>0 have an N with P(|Y_n-Y|>epsilon)<delta for all n≥N.

Main derivation

P(X_i=0)=P(X_i=1)=1/2

\mu=1/2,\quad E[X_i^2]=1/2

\sigma^2=E[X_i^2]-\mu^2

\sigma^2=1/2-1/4=1/4

\bar X_n=\frac1n\sum_{i=1}^nX_i

E[\bar X_n]=\frac1n\sum_{i=1}^n\mu=1/2

i\ne j,\quad E[X_iX_j]=E[X_i]E[X_j]

\operatorname{Cov}(X_i,X_j)=0

\operatorname{Var}(\bar X_n)=n^{-2}\sum_{i=1}^n\operatorname{Var}(X_i)

\operatorname{Var}(\bar X_n)=1/(4n)

\varepsilon>0

P(|\bar X_n-1/2|>\varepsilon)\le\frac1{4n\varepsilon^2}

\varepsilon,\delta>0,\quad N=\lfloor1/(4\varepsilon^2\delta)\rfloor+1

n\ge N\implies P(|\bar X_n-1/2|>\varepsilon)<\delta

\forall\varepsilon,\delta>0\ \exists N\ \forall n\ge N

\bar X_n\xrightarrow{P}1/2

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — It is neither an a.s. proof nor a finite-n normal equality.
• S07 · premise_audit · G1, S06, work — Epsilon and delta are arbitrary first; N may depend on them.
• S06 · premise_audit · G1, S05, work — Epsilon is positive and the mean and variance concern the same n.
• S05 · premise_audit · G1, S04, work — Do not lose the squared scaling denominator.
• S04 · premise_audit · G1, S03, work — Identical distribution alone does not ensure this equation.
• S03 · premise_audit · G1, S02, work — The denominator requires n≥1.
• S02 · premise_audit · G1, S01, work — This theorem permits zero variance; the present one is 1/4.
• S01 · premise_audit · G1, work — The index i is arbitrary, not just1.

Error controls and negative controls

Use of independence

Use independence to eliminate cross covariances; do not replace it with identical marginals.

Strict tail

Choose floor of the critical value plus one to ensure<delta.

• Equal marginals eliminate covariance. → reject — Independence is needed; repeating one variable does not average away its variance.
• The weak law proves every path converges. → reject — The conclusion here is probabilistic, not pathwise.
• A finite sample looking IID suffices. → reject — The premise covers the whole infinite sequence.

Result and limitations

Under the declared infinite IID premise, the sample mean converges in probability to 1/2.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: f11fbf0a78e4a6b10cda16b42c750644f96091f304633ea15fc9094c72cf48df
• original source SHA-256: c8293d0ee67c4f711dcf6c7b3075e1f23785264166b0253188eec000c08fc602
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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