The class of extended Pólya functions Ω = {φ: φ is a continuous real valued real function, φ(-t) = φ(t) ≤ φ(0) ∈ [0,1], límt→∞ φ(t) = c ∈ [0,1] and φ(|t|) is convex} is a convex set. Its extreme points are identified, and using Choquet's theorem it is shown that φ ∈ Ω has an integral representation of the form φ(|t|) = ∫0 ∞ 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) = (2π)-1(x/2)-2sin2(x/2), we conclude that φ is the characteristic function of the absolutely continuous random variable Z = XY, X and Y independent. Hence, any φ ∈ Ω is a characteristic function. This proof sheds an interesting light upon Pólya's sufficient condition for a given function to be a characteristic function.
@article{urn:eudml:doc:40750,
title = {A note on P\'olya's theorem.},
journal = {Trabajos de Estad\'\i stica e Investigaci\'on Operativa},
volume = {35},
year = {1984},
pages = {104-111},
zbl = {0733.60034},
mrnumber = {MR0829915},
language = {en},
url = {http://dml.mathdoc.fr/item/urn:eudml:doc:40750}
}
Pestana, Dinis. A note on Pólya's theorem.. Trabajos de Estadística e Investigación Operativa, Tome 35 (1984) pp. 104-111. http://gdmltest.u-ga.fr/item/urn:eudml:doc:40750/