A note on Pólya’s theorem

Abstract

The class of extended Polya functions O = {f: f is a continuous real valued real function, f(-t) = f(t) = f(0) I [0,1], limt?8 f(t) = c I [0,1] and f(|t|) is convex} is a convex set. Its extreme points are identified, and using Choquet's theorem it is shown that f I O has an integral representation of the form f(|t|) = ?08 max{0, 1-|t|y} dG(y), where G is the distribution function of some random variable Y. As on the other hand max{0, 1-|t|y} is the characteristic function of an absolutely continuous random variable X with probability density function f(x) = (2p)-1(x/2)-2sin2(x/2), we conclude that f is the characteristic function of the absolutely continuous random variable Z = XY, X and Y independent. Hence, any f I O is a characteristic function. This proof sheds an interesting light upon Polya's sufficient condition for a given function to be a characteristic function