Convergence in distribution

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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.


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:
This article was adapted from an original article by V.I. Bityutskov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article