Almost-sure convergence

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

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 — The variables are measurable functions on a common space.
• Sequence — The family is time-indexed.
• Probability on finite sample spaces — The exceptional set has probability zero.
• Sigma-algebras and measure spaces — The finite probability space is shared.
• 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.