Lp convergence

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

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 probability space is shared.
• Expectation in finite models — The criterion uses p-moments.
• Absolute value — Errors are taken in absolute value.
• Sequence limits and exact tail cutoffs — The moment tends to zero.
• Natural-number powers and principal square roots — The error is raised to p.
• Sigma-algebras and measure spaces — 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.