Namespaces
Variants
Actions

Convergence in probability

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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.


Comments

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: http://encyclopediaofmath.org/index.php?title=Convergence_in_probability&oldid=41768
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article