# Weak convergence of probability measures

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The general setting for weak convergence of probability measures is that of a complete separable metric space (cf. also Complete space; Separable space), being the metric, with probability measures , defined on the Borel sets of . It is said that converges weakly to in if for every bounded continuous function on one has as . If random elements , taking values in are such that the distribution of is , one writes , and says that converges in distribution to if converges weakly to (cf. also Convergence in distribution).
The metric spaces in most common use in probability are , -dimensional Euclidean space, , the space of continuous functions on , and , the space of functions on which are right continuous with left-hand limits.
Weak convergence in a suitably rich metric space is of considerably greater use than that in Euclidean space. This is because a wide variety of results on convergence in distribution on can be derived from it with the aid of the continuous mapping theorem, which states that if in and the mapping is continuous (or at least is measurable and , where is the set of discontinuities of ), then . In many applications the limit random element is Brownian motion, which has continuous paths with probability one.
One of the most fundamental weak convergence results is Donsker's theorem for sums , , of independent and identically-distributed random variables with , . This can be framed in by setting and , , where denotes the integer part of . Then Donsker's theorem asserts that , where is standard Brownian motion. Application of the continuous mapping theorem then readily provides convergence-in-distribution results for functionals such as , , , and , where is the indicator function and if and otherwise.