Chemistry - 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 F_n 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. (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(X_n) → 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. P_n 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

Categories: Probability theory | Theorems