---
tag: 0SVN6X695K
kind: statement
role: theorem
name: Almost sure convergence implies convergence in probability
msc: [60F15]
uses: [04P7D63W8P, 0EFDDXTVD6, 0X8HBKCSYB]
sources:
  - url: https://en.wikipedia.org/wiki/Convergence_of_random_variables
    locator: Properties
---

## Statement

Let $X, X_1, X_2, \dots$ be real random variables on one probability space.
If $X_n \to X$ [[almost surely|04P7D63W8P]], then $X_n \to X$ [[in probability|0EFDDXTVD6]].

## Remarks

Proof: fix $\varepsilon > 0$ and let $A_n = \bigcup_{m \ge n} \{|X_m - X| > \varepsilon\}$. The events $A_n$ decrease, and their intersection has probability $0$ by almost sure convergence.
By the [[continuity of probability|0X8HBKCSYB]], $P(A_n) \to 0$. Since $\{|X_n - X| > \varepsilon\} \subseteq A_n$, the claim follows.
The converse is false. Let $X_n$ be independent with $P(X_n = 1) = 1/n$ and $P(X_n = 0) = 1 - 1/n$. Then $X_n \to 0$ in probability.
But by the second Borel–Cantelli lemma, $X_n = 1$ for infinitely many $n$ almost surely, so $X_n$ does not converge to $0$ almost surely.

## Explanations

### L1 human

If almost every run of the experiment gives a sequence that settles down at $X$, then for large $n$ a large deviation becomes unlikely.