Convergence, almost-certain

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almost-sure convergence, convergence with probability one

Convergence of a sequence of random variables defined on a certain probability space , to a random variable , defined in the following way: (or -almost certain) if

In mathematical analysis this form of convergence is called almost-everywhere convergence. Convergence in probability follows from almost-certain convergence.


See also Convergence, types of; Weak convergence of probability measures; Distributions, convergence of.

How to Cite This Entry:
Convergence, almost-certain. V.I. Bityutskov (originator), Encyclopedia of Mathematics. URL:,_almost-certain&oldid=15210
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098