# Lp convergence

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

On a common probability space with fixed integer p≥1, Lp convergence means E|X_n−X|^p tends to zero.

mko-lp-convergence · definition · v0.10.0



## Mathematical statement

This slice uses positive integer p, a complete finite atomic law and symbolic error amplitudes c_i/(n+s). Compute exact p-moments and epsilon tail bounds. A finite prefix is insufficient; unrestricted almost-sure, other-Lq or reverse probability-mode implications are not inferred without their additional theorem assumptions.





## Representative formula (exact TeX)

L(X)





## Assumptions


- All X_n and X share one complete finite probability space with exact rational masses.
- p is a positive integer and each p-moment is finite.





## Symbol scope


- L — Lp convergence
- X — sequence of random variables





## Intuition, definition, and misconceptions

Keep p fixed and the probability space common.

The exact moment tail holds for every n≥N.

Finite-prefix evidence and general cross-mode implications need separate theorem obligations.

Finite samples or another convergence-mode label do not establish Lp convergence without the required conditions.





## Adopted derivation guide

[An exact L-squared tail on a finite law](https://ocme.evemisslab.com/en/derivations/derivation-mko-lp-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 probability space is shared.
- [Expectation in finite models](https://ocme.evemisslab.com/en/mko/mko-expectation/) — The criterion uses p-moments.
- [Absolute value](https://ocme.evemisslab.com/en/mko/mko-absolute-value/) — Errors are taken in absolute value.
- [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/mko/mko-sequence-limit/) — The moment tends to zero.
- [Natural-number powers and principal square roots](https://ocme.evemisslab.com/en/mko/mko-power-root/) — The error is raised to p.
- [Sigma-algebras and measure spaces](https://ocme.evemisslab.com/en/mko/mko-sigma-algebra-measure-space/) — All variables use the same law.





## Verification boundary

Only finite atomic models, integer p and symbolic rational amplitudes 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 convergence-mode implication theorems.





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