Helly–Bray theorem

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In probability theory, the Helly–Bray theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions. The first eponym is Eduard Helly.

Let F and F1, F2, ... be cumulative distribution functions on the real line. 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: R → R, where 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, ... be probability measures on some set S. Then 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 measures (see Billingsley, 1999, p. 3).

[edit] References

  1. Patrick Billingsley (1999). Convergence of Probability Measures, 2nd ed.. John Wiley & Sons, New York. ISBN 0-471-19745-9.

This article incorporates material from Helly–Bray theorem on PlanetMath, which is licensed under the GFDL.

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