# Central limit theorem for finite-variance IID sequences

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

An infinite IID sequence with common finite positive variance has centered sums scaled by sigma*sqrt(n) converging in distribution to the standard normal law.

mko-central-limit-theorem-finite-variance-iid · theorem · v0.10.0



## 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

[CLT qualification and a finite lattice audit](https://ocme.evemisslab.com/en/derivations/derivation-mko-central-limit-theorem-finite-variance-iid-01/)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](https://ocme.evemisslab.com/en/mko/mko-iid-sequence/) — Retain the infinite IID process premise.
- [Convergence in distribution](https://ocme.evemisslab.com/en/mko/mko-convergence-in-distribution/) — Specify the theorem conclusion.
- [Expectation in finite models](https://ocme.evemisslab.com/en/mko/mko-expectation/) — Use the common mean.
- [Variance in finite probability models](https://ocme.evemisslab.com/en/mko/mko-variance/) — Use positive finite common variance.
- [Standard normal distribution](https://ocme.evemisslab.com/en/mko/mko-standard-normal-distribution/) — Specify the limit law and CDF.
- [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/mko/mko-sigma-algebra-measure-space/) — Retain the infinite-process and distribution background.
- [Binomial theorem](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/central-limit-theorem-finite-variance-iid.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-central-limit-theorem-finite-variance-iid/index.md)