# Almost-sure convergence

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

Random variables on the same probability space converge almost surely when they converge pointwise outside an exceptional set of probability zero.

mko-almost-sure-convergence · definition · v0.10.0



## Mathematical statement

This slice uses a complete finite atomic probability space with a symbolic eventual certificate for each atom. Every positive-mass atom must converge, while a zero-mass failure may be ignored. A finite prefix does not certify a full tail; the warning that convergence in probability alone is insufficient refers to the general convergence-mode implication, not a denial of additional finite-atom results.





## Representative formula (exact TeX)

A(X)





## Assumptions


- All X_n and X use the same complete finite sample space and exact rational masses.
- An eventual certificate gives each positive atom a settling index or a universal tail formula.





## Symbol scope


- A — almost-sure-convergence record
- X — sequence of random variables





## Intuition, definition, and misconceptions

Check the declared tail convergence on each positive-mass atom.

A zero-mass failure affects pointwise convergence there, not the almost-sure conclusion.

A finite prefix is insufficient, and in general convergence in probability does not imply almost-sure convergence. The common finite-atom setting here supplies additional structure and is checked through its explicit atomwise certificates.

Finite observations do not establish an infinite tail, and a convergence-mode name alone does not supply the required explicit certificate or justify a general converse.





## Adopted derivation guide

[Atomwise tails and a null exception](https://ocme.evemisslab.com/en/derivations/derivation-mko-almost-sure-convergence-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-random-variable/) — The variables are measurable functions on a common space.
- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — The family is time-indexed.
- [Probability on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-probability/) — The exceptional set has probability zero.
- [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/mko/mko-sigma-algebra-measure-space/) — The finite probability space is shared.
- [Convergence in probability](https://ocme.evemisslab.com/en/mko/mko-convergence-in-probability/) — Distinguish the weaker convergence mode in general.





## Verification boundary

Only complete finite atomic models with explicit universal-tail certificates are handled.

not_formalized

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





## Next obligation

Add Lp convergence and relations between convergence modes.





## 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/almost-sure-convergence.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-almost-sure-convergence/index.md)