Weak law of large numbers

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

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 — Independence and the common law are premises.
• Expectation in finite models — The common mean is fixed.
• Variance in finite probability models — A finite variance gives the average-variance identity.
• 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.