Convergence in probability

Mathematical statement

All X_n and X must belong to the same probability space. Retain the tail obligations: for every epsilon>0 and every delta>0 there exists N such that every n≥N has deviation probability below delta. This slice uses an explicitly declared c/(n+s) deviation-probability profile on a valid probability domain, or an eventual-equality certificate. A finite prefix does not establish convergence.

Representative formula (exact TeX)

P(X)

Assumptions

• The symbolic c/(n+s) profile has positive integers c,s and starts only where its probability is at most 1.
• Convergence in probability is not conflated with almost-sure, L^p or distributional convergence.

Symbol scope

• P — convergence-in-probability record
• X — common limiting random variable

Intuition, definition, and misconceptions

Every positive epsilon requires control of the corresponding deviation event.

The c/(n+s) profile gives an explicit N for any positive delta on its valid starting domain.

Finite prefixes support only an audit; eventual equality provides a different universal-tail certificate.

Finitely many decreasing error probabilities do not establish convergence, and convergence in distribution is not the same notion.

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

• Finite joint distribution — A common finite probability space supplies the joint setting.
• Real random variables on finite sample spaces — X_n and X are variables on the same space.
• Probability on finite sample spaces — Deviation events have probabilities.
• Sequence — The variables are indexed by n.
• Convergence in a metric space — Real-valued deviations use distance.

Verification boundary

Only explicitly declared symbolic tails or eventual equality are accepted; no almost-sure, Lp or CLT conclusion follows. The active zero-indexed prefix uses the owner-adopted minimal-shift revision of 2026-10-17 so every displayed value is in [0,1]; the original out-of-domain prefix remains preserved as history.

not_formalized

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

Next obligation

Add convergence in distribution and the weak law of large numbers.

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.