Strong law of large numbers for finite-valued IID sequences

Mathematical statement

Assume an explicit infinite IID process and a normalized finite common marginal law with mean mu. The strong-law conclusion is P(lim_(n→infinity) n^(-1)sum_(i=1)^n X_i=mu)=1. Exact families recalculate the marginal mean, a fixed prefix sample mean and a complete fixed-n deviation-event probability. They expressly do not certify the probability-one limit. The infinite_iid certificate token is a declared process premise, not verification of infinite independence. Zero variance is allowed. Non-IID and nonfinite support are outside this finite-valued theorem scope, not counterexamples to every SLLN; a finite prefix does not establish a limit and a different target is not the common mean.

Representative formula (exact TeX)

S(X,n,e)=b

Assumptions

• The declared process premise covers infinitely many indices, independence and the common finite law.
• The finite marginal carrier has exact nonnegative probabilities summing to1.

Symbol scope

• S — strong-law operator
• X — infinite finite-valued IID sequence
• n — prefix length
• e — positive deviation
• b — almost-sure convergence statement

Intuition, definition, and misconceptions

A finite marginal law does not replace the infinite IID process premise.

Prefix means and deviation probabilities are fixed-n audits only.

An outside-scope input is not automatically a counterexample to a general strong law.

A few close prefix means do not prove almost-sure convergence, and nonfinite support does not force every SLLN to fail.

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 premise.
• Almost-sure convergence — Specify the probability-one limit.
• Expectation in finite models — Use the common mean.
• Sigma-algebras and measure spaces — Retain the almost-sure event background.
• Variance in finite probability models — Use the finite-law moment background.

Verification boundary

Finite prefix enumeration does not prove a probability-one limit; this conclusion is conditional on the declared process and named theorem.

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 integrable IID SLLN, Borel–Cantelli arguments and ergodic boundaries.

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.