Central limit theorem for finite-variance IID sequences

Mathematical statement

Assume an infinite IID process with common mean mu and0<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 and1, 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.

Representative formula (exact TeX)

C(X,n,z)=b

Assumptions

• The process is an explicitly declared infinite IID sequence, not inferred from a finite sample.
• The common law has finite mean and strictly positive finite variance.
• Center by n*mu and scale by sigma*sqrt(n), not n*sigma or an uncentered sum.

Symbol scope

• C — central-limit operator
• X — infinite IID sequence
• n — sample size
• z — normal-CDF threshold
• b — distributional-convergence statement

Intuition, definition, and misconceptions

Qualify the infinite IID process and moments before invoking the theorem.

Fixed-n standardized moments and binomial lattices are reproducible finite calculations, not convergence proofs.

Separate zero variance, non-IID, wrong scaling and finite-histogram boundaries.

One bell-shaped histogram does not prove the CLT, and sqrt(n) cannot be replaced by n.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Independent identically distributed sequence — Retain the infinite IID process premise.
• Convergence in distribution — Specify the theorem conclusion.
• Expectation in finite models — Use the common mean.
• Variance in finite probability models — Use positive finite common variance.
• Standard normal distribution — Specify the limit law and CDF.
• Sigma-algebras and measure spaces — Retain the infinite-process and distribution background.
• Binomial theorem — Calculate exact Bernoulli sum probabilities.

Verification boundary

Finite laws and lattice tables certify only the specified n; they do not prove the general distributional limit. The infinite process remains a declared premise, and the general theorem is not formalized here.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add Lindeberg conditions, Berry–Esseen bounds and the multivariate CLT.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.