Difference between revisions of "Convergence in probability"
From Encyclopedia of Mathematics
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| − | Convergence of a sequence of random | + | Convergence of a sequence of [[random variable]]s $X_1,X_2,\ldots$ 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]]. | |
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| + | ====Comments==== | ||
| + | See also [[Weak convergence of probability measures]]; [[Convergence, types of]]; [[Distributions, convergence of]]. | ||
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Latest revision as of 19:30, 1 September 2017
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
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