CorollaryStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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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.
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Kolmogorov strong law for independent uniformly bounded variances

Statement

Independent square-integrable real $(X_{n})_{n \geq 1}$ with $C = \sup_{n} Var (X_{n}) < \infty$ satisfy $(S_{n} - E S_{n}) / n \to 0$ almost surely.

Facts & Assumptions

[F1]

Strong law under summable normalized variances: Let $(X_{n})_{n \geq 1}$ be independent square-integrable real random variables. Let $0 < b_{n}$ be deterministic and nondecreasing with $b_{n} \to \infty$ . If $\sum_{n \geq 1} \frac{Var (X_{n})}{b_{n}^{2}} < \infty,$ then $\frac{1}{b_{n}} \sum_{k = 1}^{n} (X_{k} - E X_{k}) ⟶ 0 almost surely .$ In particular, for IID centered square-integrable variables and any $ε > 0$ , $S_{n} / [\sqrt{n} (\log n)^{1 / 2 + ε}] \to 0$ almost surely (the displayed normalization is used for $n \geq 2$ ).

Proof

Given: The objects, hypotheses and definitions in the statement. Its conclusions are to be established below.

1.1

For $n \geq 2$ , $n^{- 2} \leq (n - 1)^{- 1} - n^{- 1}$ . Therefore $\sum_{n = 1}^{N} Var (X_{n}) / n^{2} \leq C (2 - 1 / N) \leq 2 C$ ; these nonnegative partial sums have a finite supremum, so the variance series converges.

givenalgebra

2.1

The normalizers $b_{n} = n$ are positive, nondecreasing and tend to infinity. The independence and square-integrability are given, and step 1.1 verifies the summability hypothesis of F1. Its almost-sure conclusion is exactly the stated centered law.

F1step 1.1∎

Depends on

Used by

Iid finite variance strong law Corollary

Dependency tree · two levels

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Sources

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