DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

$L^{p}$ convergence for random variables

Definition

Let $1 \leq p \leq \infty$ . For real random variables whose classes lie in $L^{p} (P)$ as defined by The space $L^{p} (μ)$ as the quotient by null functions, write $X_{n} \to X$ in $L^{p}$ when $∥ [X_{n}] - [X] ∥_{L^{p} (P)} ⟶ 0.$ For $p < \infty$ , this norm is $∥ [X_{n}] - [X] ∥_{L^{p} (P)} = (E ∣ X_{n} - X ∣^{p})^{1 / p};$ for $p = \infty$ , it is the essential-supremum norm. Thus the assertion concerns almost-everywhere equivalence classes, not chosen representatives.

Depends on

Used by

Almost-sure convergence need not imply Lᵖ convergence Counterexample
Convergence in probability need not imply Lᵖ convergence Counterexample
Lᵖ convergence need not imply almost-sure convergence Counterexample
Lᵖ convergence need not imply L^q convergence when p<q Counterexample
Dominated convergence in Lᵖ Theorem
L^q convergence implies Lᵖ convergence on a probability space Theorem
L¹ convergence implies uniform integrability Theorem
Lᵖ convergence implies convergence in probability Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

S. Roch, Lecture 3: Modes of convergence, Definition 3.1 (standard reference, not scraped)

LpLp convergence for random variables