LemmaStatement: 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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Iid linear truncation occurs only finitely often

Statement

For identically distributed integrable real $(X_{n})$ , put $Y_{n} = X_{n} 1_{{∣ X_{n} ∣ \leq n}}$ . Almost surely $Y_{n} = X_{n}$ for all sufficiently large $n$ . Consequently $n^{- 1} \sum_{k = 1}^{n} (X_{k} - Y_{k}) \to 0$ . Independence is unnecessary.

Facts & Assumptions

[F1]

Tail sum integrability equivalence: For a measurable $X : Ω \to [0, \infty]$ on a probability space, $\sum_{n \geq 1} P (X > n) \leq E X \leq 1 + \sum_{n \geq 1} P (X > n)$ . Thus $E X < \infty$ if and only if the tail series is finite.

[F2]

First Borel-Cantelli lemma for events: Let $(A_{n})_{n \in N}$ be events in a probability space. If $\sum_{n = 0}^{\infty} P (A_{n}) < + \infty,$ then $P (A_{n} i . o .) = 0.$

No independence hypothesis is needed.

Proof

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

1.1F1

The common law gives $P (∣ X_{n} ∣ > n) = P (∣ X_{1} ∣ > n)$ . By F1 their sum is at most $E ∣ X_{1} ∣ < \infty$ .

2.1F2∎

Apply F2 to these events. Outside their null limsup, a finite $N (ω)$ bounds all exceptional indices and $X_{k} - Y_{k} = 0$ for $k > N (ω)$ . Hence for $n > N (ω)$ the numerator is the fixed finite real sum $\sum_{k \leq N (ω)} (X_{k} - Y_{k})$ , and its quotient by $n$ tends to zero.

Depends on

Used by

Etemadi strong law for pairwise independent iid variables Theorem
Kolmogorov iid l1 strong law Theorem

Dependency tree · two levels

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Sources

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