Difference between revisions of "Convergence in distribution"
From Encyclopedia of Mathematics
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Convergence of a sequence of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026050/c0260501.png" /> defined on a certain probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026050/c0260502.png" />, to a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026050/c0260503.png" />, defined in the following way: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026050/c0260504.png" /> if | Convergence of a sequence of random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026050/c0260501.png" /> defined on a certain probability space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026050/c0260502.png" />, to a random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026050/c0260503.png" />, defined in the following way: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026050/c0260504.png" /> if | ||
Latest revision as of 16:40, 6 February 2012
2020 Mathematics Subject Classification: Primary: 60B10 [MSN][ZBL]
Convergence of a sequence of random variables 
defined on a certain probability space 
, to a random variable 
, defined in the following way: 
if
 | (*) |
for any bounded continuous function 
. This form of convergence is so called because condition (*) is equivalent to the convergence of the distribution functions 
to the distribution function 
at every point 
at which 
is continuous.
Comments
See also Convergence, types of; Distributions, convergence of.
This is special terminology for real-valued random variables for what is generally known as weak convergence of probability measures (same definition as in (*), but with 
, 
taking values in possibly more general spaces).
How to Cite This Entry:
Convergence in distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convergence_in_distribution&oldid=19094
Convergence in distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Convergence_in_distribution&oldid=19094
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article