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# Helly-Bray theorem

The 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

$\int_\mathbb{R} g(x)\,dF_n(x) \rightarrow \int_\mathbb{R} g(x)\,dF(x), \quad n\rightarrow\infty,$

for each bounded, continuous function g: RR. (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

$\int_S g \,dP_n \rightarrow \int_S g \,dP, \quad n\rightarrow\infty,$

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

Last updated: 06-04-2005 17:31:07
03-10-2013 05:06:04