# Weak law of large numbers

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

For IID variables with finite variance, a Chebyshev bound shows the sample mean converges in probability to the common mean.

mko-weak-law-large-numbers · theorem · v0.10.0



## Mathematical statement

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.





## Representative formula (exact TeX)

\bar{X}_n\xrightarrow{\mathrm{P}}\mu





## Assumptions


- IID concerns every finite subfamily of the whole sequence, not observations of a finite sample.
- The common mean and variance are finite, and epsilon and delta are positive rationals.





## Symbol scope


- \bar{X}_n — sample mean
- \mu — common mean
- X — iid sequence
- n — positive sample size





## Intuition, definition, and misconceptions

Independence reduces the variance of the average to sigma_squared/n.

The Chebyshev bound decreases to zero and supplies a witness for every later n.

The conclusion is convergence in probability, not almost-sure convergence, a CLT or finite-n equality with the limit.

Notation candidate: bar(X)_n=(X_1+...+X_n)/n and μ is the common mean; upright P on the arrow explicitly means convergence in probability. Existing whole-sequence IID and finite-variance premises remain; this is neither convergence of individual coordinates nor an almost-sure or finite-simulation proof.

A few simulations or one finite sample path do not prove the law, and the WLLN is not the CLT.





## Adopted derivation guide

[A Chebyshev tail for fair Bernoulli averages](https://ocme.evemisslab.com/en/derivations/derivation-mko-weak-law-large-numbers-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/) — Independence and the common law are premises.
- [Expectation in finite models](https://ocme.evemisslab.com/en/mko/mko-expectation/) — The common mean is fixed.
- [Variance in finite probability models](https://ocme.evemisslab.com/en/mko/mko-variance/) — A finite variance gives the average-variance identity.
- [Convergence in probability](https://ocme.evemisslab.com/en/mko/mko-convergence-in-probability/) — The target is an epsilon probability tail.





## Verification boundary

This is the finite-variance Chebyshev version, not the most general WLLN or a strong law.

not_formalized

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





## Next obligation

Add the strong law and central limit theorem with their independent premises.





## 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/source13-20261018-weak-law-large-numbers.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-weak-law-large-numbers/index.md)