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, convergence in probability, convergence in mean and convergence in distribution. 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.