DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

Strong law of large numbers for a sequence

Definition

Let $(X_{n})_{n \geq 1}$ be integrable real random variables on one probability space, with $S_{n} = \sum_{k = 1}^{n} X_{k}$ . The centered strong law means $(S_{n} - E S_{n}) / n \to 0$ almost surely. If the variables have a common law, their finite expectations equal $μ = E X_{1}$ , so this is equivalent to $S_{n} / n \to μ$ almost surely. Independence is not part of this definition.

Depends on

Used by

Kolmogorov strong law for independent uniformly bounded variances Corollary
Identical distribution without independence can defeat the mean law Counterexample
Weak law does not imply strong law Counterexample

Dependency tree · two levels

10 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

Durrett, §§2.4–2.5, pp. 76–87 (standard reference, not scraped)