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Uniform convergence of probability measures on classes of functions. (English) Zbl 0821.60002

Summary: Let \(P_ n\), \(P\) be probabilities, and \(\mathbf F\), \({\mathbf F}^*\) be collections of real functions. Simple conditions are derived under which the simple convergence of \(\int f(x) P_ n (dx)\) to \(\int f(x) P(dx)\) for every \(f\) in \({\mathbf F}^*\) implies uniform convergence over \({\mathbf F} :\sup_{f \in F} | \int f(x) P_ n (dx) - \int f(x) P(dx) |\) converges to 0. Several examples are discussed, some historical and some new.

MSC:

60B10 Convergence of probability measures
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