CorollaryStatement: AI-adaptedProof: AI-generatedPipeline-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.

Kolmogorov two-series sufficiency

Statement

Let $(X_{n})_{n \geq 1}$ be independent square-integrable real random variables. If $\sum_{n \geq 1} E X_{n}$ converges in $R$ and $\sum_{n \geq 1} Var (X_{n}) < \infty$ , then $\sum_{n \geq 1} X_{n}$ converges almost surely and in $L^{2}$ .

Facts & Assumptions

[F1]

Kolmogorov convergence criterion: For independent centered square-integrable real random variables $(X_{n})_{n \geq 1}$ , if $\sum_{n \geq 1} Var (X_{n}) < \infty$ , then $\sum_{n \geq 1} X_{n}$ converges almost surely and in $L^{2}$ to the same finite real random variable.

[F2]

Measurable coordinatewise functions preserve independence: Let $(X_{i})_{i \in I}$ be an independent family of random elements $X_{i} : (Ω, F, P) \to (S_{i}, Σ_{i})$ . For each $i$ , let $g_{i} : (S_{i}, Σ_{i}) \to (T_{i}, T_{i})$ be measurable. Then the family $(g_{i} \circ X_{i})_{i \in I}$ is independent.

[F3]

Variance and covariance identities for random variables: Let $X, Y$ be square-integrable real random variables on one probability space. Then $Var (X) = E [X^{2}] - E [X]^{2},$ $Cov (X, Y) = E [X Y] - E [X] E [Y] .$ Moreover, covariance is symmetric and bilinear on finite linear combinations. On finite full-power-set probability spaces these formulas reduce to the published finite identities.

Proof

Given: The objects and hypotheses of the statement.

1.1

Set $Y_{n} = X_{n} - E X_{n}$ . These are independent centered square-integrable variables, and $Var (Y_{n}) = Var (X_{n})$ by covariance bilinearity. The convergence criterion supplies an almost-sure and $L^{2}$ limit $Y$ for their series.

F2F3F1given

2.1

Let $a = \sum_{n} E X_{n}$ . The identity $\sum_{k = 1}^{n} X_{k} = \sum_{k = 1}^{n} Y_{k} + \sum_{k = 1}^{n} E X_{k}$ gives pointwise convergence to $Y + a$ on the same event. Its $L^{2}$ error is at most the centered $L^{2}$ error plus $∣ \sum_{k = 1}^{n} E X_{k} - a ∣$ , which tends to zero. This includes zero variances and conditionally convergent deterministic means.

step 1.1givenalgebra∎

Depends on

Used by

Kolmogorov three-series theorem Theorem

Dependency tree · two levels

23 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

Theorem 3.11, p. 66 (standard reference, not scraped)