In probability theory, the Helly–Bray theorem relates the weak convergence of cumulative distribution functions to the convergence of expectations of certain measurable functions. It is named after Eduard Helly and Hubert Evelyn Bray.
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
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 measures (see Billingsley, 1999, p. 3).
- Patrick Billingsley (1999). Convergence of Probability Measures, 2nd ed. John Wiley & Sons, New York. ISBN 0-471-19745-9.