Convergence in probability

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2020 Mathematics Subject Classification: Primary: 60-01 Secondary: 28A20 [MSN][ZBL]

Convergence of a sequence of random variables $X_1,X_2,\ldots$ defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, to a random variable $X$, defined in the following way: $X_n \stackrel{\mathrm{P}}{\rightarrow} X$ if for any $\epsilon > 0$, $$ \mathbb{P}\{ |X_n-X| > \epsilon \} \rightarrow 0 \ \ \text{as}\ \ n \rightarrow \infty \ . $$

In mathematical analysis, this form of convergence is called convergence in measure. Convergence in probability implies convergence in distribution.


See also Weak convergence of probability measures; Convergence, types of; Distributions, convergence of.

How to Cite This Entry:
Convergence in probability. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article