TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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An almost-surely convergent subsequence from convergence in probability

Statement

If $X_{n} \to X$ in probability, then some subsequence $X_{n_{k}}$ converges to $X$ almost surely.

Facts & Assumptions

Given: $X_{n} \to X$ in probability.

[L1]

Convergence in probability makes each fixed-threshold error probability eventually arbitrarily small (Convergence in probability).

[L2]

A summable sequence of event probabilities gives only finitely many of those events almost surely (First Borel-Cantelli lemma for events).

Proof

technique · constructive

1.1

Recursively choose $n_{k} > n_{k - 1}$ least subject to the following bound. [L1, construct] $P (∣ X_{n_{k}} - X ∣ > 2^{- k}) < 2^{- k}$ ; [L1] makes every choice possible. Put $A_{k} = {∣ X_{n_{k}} - X ∣ > 2^{- k}}$ .

L1construct

2.1

The bounds in step 1.1 are summable, so [L2] applies. [step 1.1, L2, discharge-construct] It says only finitely many $A_{k}$ occur almost surely. Hence $∣ X_{n_{k}} - X ∣ \leq 2^{- k}$ eventually almost surely, so $X_{n_{k}} \to X$ almost surely.

step 1.1L2discharge-construct∎

Depends on

Used by

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

S. Roch, Lecture 3: Modes of convergence, Theorem 3.12 (standard reference, not scraped)