# Convergence in probability

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

X_n converges in probability to X when, for every epsilon>0, the probability of deviation greater than epsilon tends to zero.

mko-convergence-in-probability · definition · v0.10.0



## 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

[A universal tail certificate from eventual equality](https://ocme.evemisslab.com/en/derivations/derivation-mko-convergence-in-probability-01/)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](https://ocme.evemisslab.com/en/mko/mko-finite-joint-distribution/) — A common finite probability space supplies the joint setting.
- [Real random variables on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-random-variable/) — X_n and X are variables on the same space.
- [Probability on finite sample spaces](https://ocme.evemisslab.com/en/mko/mko-probability/) — Deviation events have probabilities.
- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — The variables are indexed by n.
- [Convergence in a metric space](https://ocme.evemisslab.com/en/mko/mko-metric-convergence/) — 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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/convergence-in-probability.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-convergence-in-probability/index.md)