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Let F and F1, F2, ... be distribution functions. The Helly-Bray theorem states that if Fn converges weakly to F, then
Note that if X and X1, X2, ... are random variables corresponding to these distribution functions, then the Helly-Bray theorem does not imply that E(Xn) → E(X), since g(x) = x is not a bounded function.
The more general theorem above is sometimes taken as defining weak convergence of probability measures (see Billingsley, 1999, p. 3).
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