CLT qualification and a finite lattice audit

Goal and scope

Separate finite-n moments from the named CLT application.

The CLT is an explicit named theorem prerequisite, not proved here.

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.

• S_n — partial sum
• Z_n — correct standardization
• Φ — standard-normal CDF

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• mko_definition · cited — Assume an infinite IID process with common mean mu and 0<sigma^2<infinity. For S_n=sum_(i=1)^n X_i define Z_n=(S_n−n*mu)/(sigma*sqrt(n)). Then P(Z_n≤z) tends to Phi(z) at every continuity point of the standard normal CDF. The exact families use a finite common marginal law, recalculate its moments, the fixed-n standardized moments0 and 1, and a complete Bernoulli sum lattice. A symbolic branch conditionally invokes the named theorem under declared hypotheses. The infinite_iid token declares a premise; it does not construct or prove a whole process. A finite binomial table or histogram is not a normal approximation or convergence proof. Non-IID excludes this particular theorem; zero variance and n*sigma scaling do not qualify this normalization. · mko-central-limit-theorem-finite-variance-iid
• work · prerequisite — Explicit named theorem prerequisite: infinite IID with 0<sigma²<infinity gives convergence of centered sums divided by sigma sqrt(n) to standard normal; also assume independent variance addition and the binomial law.

Main derivation

X_i\sim\operatorname{Bernoulli}(1/2)

\mu=1/2,\quad\sigma^2=1/4\in(0,\infty)

S_n=\sum_{i=1}^nX_i

E[S_n]=n/2,\quad\operatorname{Var}(S_n)=n/4

Z_n=(S_n-n\mu)/(\sigma\sqrt n)

Z_n=\frac{S_n-n/2}{\sqrt n/2}

E[Z_n],\quad\operatorname{Var}(Z_n)

E[Z_n]=0,\quad\operatorname{Var}(Z_n)=\frac{n/4}{n/4}=1

n=4,\quad k\in\{0,1,2,3,4\}

P(S_4=k)=\binom4k/16=(1,4,6,4,1)_k/16

Z_4=S_4-2

Z_4\in\{-2,-1,0,1,2\},\quad P(Z_4=0)=3/8

\text{infinite IID},\ E[X_i]=\mu,\ 0<\sigma^2<\infty

\forall z\in\mathbb R:\lim_{n\to\infty}P(Z_n\le z)=\Phi(z)\quad\text{(named CLT)}

P(Z_4=0)=3/8,\quad P(Z=0)=0\ (Z\sim N(0,1))

Z_n\Rightarrow N(0,1),\quad Z_4\not\sim N(0,1)

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — No normal-approximation error rate or formal CLT proof is supplied.
• S07 · premise_audit · G1, S06, work — This limit comes from the named theorem, not the five-point table.
• S06 · premise_audit · G1, S05, work — The standard normal is atomless, ruling out exact equality of laws.
• S05 · premise_audit · G1, S04, work — The coefficients sum to 16 from the four-fold Bernoulli product.
• S04 · premise_audit · G1, S03, work — Moments0 and 1 alone do not identify a normal law.
• S03 · premise_audit · G1, S02, work — It is not n/2, and centering must subtract n/2.
• S02 · premise_audit · G1, S01, work — Variance addition uses the declared whole-sequence independence.
• S01 · premise_audit · G1, work — Zero variance invalidates normalization, so positivity cannot be omitted.

Error controls and negative controls

Premise layer

Declare infinite IID as a premise, not a process-construction proof.

Nondegeneracy and scale

Use sigma²=1/4>0 and denominator sigma sqrt(n); retain finite-n discreteness.

• The five-point n=4 law is normal. → reject — It has mass 3/8 at 0; a normal law has zero singleton mass.
• Dividing by n sigma also gives variance 1. → reject — It gives variance 1/n instead of 1.
• Checking the lattice table proves the CLT. → reject — Finite-n data does not establish the infinite CDF limit; that uses the named theorem.

Result and limitations

The named CLT applies, so Z_n⇒N(0,1); the n=4 lattice neither proves the limit nor equals a normal law.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 3b87490489b4be576acff8199037a9cb5b933e355624ff89027db6b289b31c18
• original source SHA-256: 45953c182abd08cc1f8022fde9a4c32399f9ee79d64b005a7d3c12e73d5f5add
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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