# Strong law of large numbers for finite-valued IID sequences

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

The sample means of a finite-valued infinite IID sequence converge almost surely to the common expectation.

mko-strong-law-large-numbers-finite-valued-iid · theorem · v0.10.0



## 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

[Finite-valued IID strong-law qualification and fixed-n boundaries](https://ocme.evemisslab.com/en/derivations/derivation-mko-strong-law-large-numbers-finite-valued-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 premise.
- [Almost-sure convergence](https://ocme.evemisslab.com/en/mko/mko-almost-sure-convergence/) — Specify the probability-one limit.
- [Expectation in finite models](https://ocme.evemisslab.com/en/mko/mko-expectation/) — Use the common mean.
- [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/mko/mko-sigma-algebra-measure-space/) — Retain the almost-sure event background.
- [Variance in finite probability models](https://ocme.evemisslab.com/en/mko/mko-variance/) — 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/strong-law-large-numbers-finite-valued-iid.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-strong-law-large-numbers-finite-valued-iid/index.md)