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

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Almost-sure convergence implies convergence in probability

Statement

If $X_{n} \to X$ almost surely, then $X_{n} \to X$ in probability.

Facts & Assumptions

Given: $X_{n} \to X$ almost surely.

[L1]

Real random variables are measurable, so differences, absolute values, threshold events, and their indicators are measurable (Random elements and real random variables).

[L2]

Dominated convergence sends an almost-everywhere convergent integrable sequence with one integrable majorant to convergence of integrals (Dominated convergence).

Proof

technique · direct

1.1givenL1

Fix $ε > 0$ and set $I_{n} = 1_{{∣ X_{n} - X ∣ > ε}}$ . [given, L1] By [L1] these are measurable; the hypothesis gives $I_{n} \to 0$ almost surely, and $0 \leq I_{n} \leq 1$ .

2.1step 1.1L2∎

Apply [L2] to the indicators from step 1.1 with majorant $1$ . [step 1.1, L2] $P (∣ X_{n} - X ∣ > ε) = E I_{n} \to 0$ , which is the required probability convergence.

Depends on

Used by

Iid strong law implies the weak law Corollary
Glivenko–Cantelli classes Definition
Cauchy sequences in probability have a measurable limit Lemma
Complete convergence implication diagram Remark
Conditional IID bootstrap CLT for an empirical law Theorem
Convergence in probability and almost surely agree for independent series Theorem
Reverse martingale convergence Theorem
Subsequence characterization of convergence in probability Theorem
Uniformly integrable martingale convergence Theorem

Dependency tree · two levels

18 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)