Difference between revisions of "Convergence in probability"
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Convergence of a sequence of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026080/c0260801.png" /> defined on a probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026080/c0260802.png" />, to a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026080/c0260803.png" />, defined in the following way: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026080/c0260804.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026080/c0260805.png" />, | Convergence of a sequence of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026080/c0260801.png" /> defined on a probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026080/c0260802.png" />, to a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026080/c0260803.png" />, defined in the following way: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026080/c0260804.png" /> if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026080/c0260805.png" />, | ||
Revision as of 15:11, 7 February 2012
2020 Mathematics Subject Classification: Primary: 60-01 Secondary: 28A20 [MSN][ZBL]
Convergence of a sequence of random variables 
defined on a probability space 
, to a random variable 
, defined in the following way: 
if for any 
,
 |
In mathematical analysis, this form of convergence is called convergence in measure. Convergence in distribution follows from convergence in probability.
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=20866
Convergence in probability. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convergence_in_probability&oldid=20866
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article