Distributions, convergence of

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Weak convergence or convergence in variation, and defined as follows. A sequence of distributions (probability measures) on the Borel sets of a metric space is called weakly convergent to a distribution if


for any real-valued bounded continuous function on . Weak convergence is a basic type of convergence considered in probability theory. It is usually denoted by the sign . The following conditions are equivalent to weak convergence:

1) (*) holds for any bounded uniformly-continuous real-valued function ;

2) (*) holds for any bounded -almost-everywhere continuous real-valued function ;

3) for any closed set ;

4) for any open set ;

5) for any Borel set with , where is the boundary of ;

6) , where is the Lévy–Prokhorov metric.

Let be a class of subsets of , closed under intersection and such that every open set in is a finite or countable union of sets in . Then if for all , it follows that . If and , are the distribution functions corresponding to , respectively, then if and only if at every point where is continuous.

Let be a separable space and let be the class of real-valued bounded Borel functions on . To have uniformly over for every sequence such that , it is necessary and sufficient that:




and is the open ball of radius with centre . If the class is generated by the indicator functions of sets from some class , then conditions a) and b) lead to the condition


(when each open ball in is connected, ). If and the distribution is absolutely continuous with respect to Lebesgue measure, then if and only if uniformly over all convex Borel sets .

Let , be distributions on a metric space such that and let be a continuous -almost-everywhere measurable mapping of into a metric space . Then , where for any distribution on , the distribution is its -image on :

for any Borel set .

A family of distributions on is said to be weakly relatively compact if every sequence of elements of it contains a weakly convergent subsequence. A condition for weak relative compactness is given by Prokhorov's theorem. A family is called tight if, for any , there is a compact set such that , for all . Prokhorov's theorem now states: If is tight, then it is relatively compact; if, moreover, is separable and complete, then weak relative compactness of implies its tightness. In the case when , a family of distributions is weakly relatively compact if and only if the family of characteristic functions corresponding to is equicontinuous at zero.

Now let , be distributions on a measure space , where is a -algebra. Convergence in variation of to means uniform convergence on all sets in or, equivalently, convergence on all sets in or, equivalently, convergence of the variation

to zero. Here, and are the components in the Jordan–Hahn decomposition of the signed measure .


[1] P. Billingsley, "Convergence of probability measures" , Wiley (1968)
[2] M. Loève, "Probability theory" , Princeton Univ. Press (1963)
[3] R.N. Bhattacharya, R. Ranga Rao, "Normal approximations and asymptotic expansions" , Wiley (1976)


For more information on weak convergence see Weak convergence of probability measures; Convergence of measures.

How to Cite This Entry:
Distributions, convergence of. Encyclopedia of Mathematics. URL:,_convergence_of&oldid=11307
This article was adapted from an original article by V.V. Sazonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article