---
tag: 0FQ968YKA1
kind: concept
name: Convergence of random variables
aliases: [modes-of-convergence-of-random-variables, stochastic-convergence]
msc: [60F05, 60F15]
uses: [04P7D63W8P, 0EFDDXTVD6, 0JDAQHWA9N, 0TTA57SEJG]
sources:
  - url: https://en.wikipedia.org/wiki/Convergence_of_random_variables
    locator: lead section
same_as:
  wikidata: Q578985
  wikipedia: Convergence of random variables
---

## Summary

A sequence of random variables can converge to a limit in several different senses.
The main modes are [[almost sure convergence|04P7D63W8P]], [[convergence in probability|0EFDDXTVD6]], [[convergence in mean|0TTA57SEJG]] and [[convergence in distribution|0JDAQHWA9N]].
Almost sure convergence and convergence in mean each imply convergence in probability, and convergence in probability implies convergence in distribution. The converse implications are false in general.

## Remarks

Convergence in distribution concerns only the laws of the random variables. So the random variables can be on different probability spaces.
The law of large numbers is a statement about convergence in probability or almost surely. The central limit theorem is a statement about convergence in distribution.

## Explanations

### L1 human

For numbers, there is one meaning of "$x_n \to x$". For random variables, there are several, because a random variable has many values at once.
Some modes ask that the values themselves come close, and the weakest mode asks only that the probabilities come close.