# Convergence in distribution

From Encyclopedia of Mathematics

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

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