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

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Integrability is necessary for an iid finite mean strong law

Statement

If IID real $(X_{n})$ have $S_{n} / n$ converging almost surely to a finite, possibly random, limit $L$ , then $E ∣ X_{1} ∣ < \infty$ and $L = E X_{1}$ almost surely.

Facts & Assumptions

[F1]

Second Borel-Cantelli lemma under pairwise independence: Let $(A_{n})_{n \in N}$ be pairwise independent events with $\sum_{n = 0}^{\infty} P (A_{n}) = + \infty .$ Then $P (A_{n} i . o .) = 1.$

[F2]

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.

[F3]

Kolmogorov iid l1 strong law: For IID real $(X_{n})_{n \geq 1}$ with $E ∣ X_{1} ∣ < \infty$ , $S_{n} / n \to μ = E X_{1}$ almost surely.

Proof

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

1.1

On the given conull convergence event, for $n \geq 2$ one has $X_{n} / n = S_{n} / n - ((n - 1) / n) (S_{n - 1} / (n - 1)) \to L - L = 0$ . Therefore the events $A_{n} = {∣ X_{n} ∣ > n}$ occur only finitely often almost surely.

givenalgebra

2.1

The events $A_{n}$ are independent because each belongs to the $σ$ -algebra of its own coordinate. If $\sum_{n} P (A_{n})$ were infinite, F1 would make their limsup conull, contradicting step 1.1. The series is therefore finite, and identical laws turn it into $\sum_{n} P (∣ X_{1} ∣ > n)$ .

F1step 1.1

3.1

F2 applied to step 2.1 gives $E ∣ X_{1} ∣ < \infty$ . Now F3 gives $S_{n} / n \to E X_{1}$ almost surely. Uniqueness of a finite real limit on the intersection of the two conull events identifies L as claimed.

F2F3step 2.1∎

Depends on

Used by

Iid strong law fails at infinite absolute mean Counterexample

Dependency tree · two levels

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Sources

Durrett, Theorem 2.3.8 and §2.4 (standard reference, not scraped)