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Helly-bray TheoremThe Helly-Bray theorem in probability theory relates the weak convergence of distribution functions to the convergence of expectations of certain measurable functions. Let F and F1, F2, ... be distribution functions. The Helly-Bray theorem states that if Fn converges weakly to F, then -
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for each bounded, continuous function g: R → R. (The integrals involved are Riemann-Stieltjes integrals.) 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. In fact, a stronger and more general theorem holds. Let P and P1, P2, ... are probability measures on some set S. Pn converges weakly to P if and only if -
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for all bounded, continuous and real-valued functions on S. (The integrals in this version of the theorem are Lebesgue-Stieltjes integrals.) The more general theorem above is sometimes taken as defining weak convergence of probability measures (see Billingsley, 1999, p. 3). References
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