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
Lagged lattice covariance and variogram estimators Definition
Asymptotic unbiasedness of lattice covariance and variogram estimators Theorem
Chebyshev weak law for uncorrelated arrays Theorem
Dominated convergence in Lᵖ Theorem
Kolmogorov convergence criterion 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
Unbiasedness and L2 consistency of the spatial window average 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