---
tag: 04P7D63W8P
kind: definition
name: Almost sure convergence
msc: [60F15]
defines: 0AAXWGT5TD
uses: [0R46A1XY60, 0DKRR22AEQ]
sources:
  - url: https://en.wikipedia.org/wiki/Convergence_of_random_variables
    locator: Almost sure convergence
---

## Definition

Let $X, X_1, X_2, \dots$ be real random variables on one probability space $(\Omega, \mathcal{F}, P)$.
The sequence $(X_n)$ converges almost surely to $X$ if
$$P\left(\left\{\omega \in \Omega : \lim_{n \to \infty} X_n(\omega) = X(\omega)\right\}\right) = 1.$$
One writes $X_n \to X$ a.s. or $X_n \xrightarrow{\text{a.s.}} X$.

## Remarks

The set in the definition is an event, because the $X_n$ and $X$ are measurable.
An equivalent condition: for each $\varepsilon > 0$, $P(|X_n - X| > \varepsilon \text{ for infinitely many } n) = 0$.
The limit is unique up to equality [[almost surely|0R46A1XY60]].
The strong law of large numbers is a result about almost sure convergence.

## Explanations

### L1 human

Almost sure convergence means that, apart from a set of outcomes of probability $0$, each run of the experiment gives a convergent sequence of numbers.