# A note on Pólya's theorem.

• Volume: 35, Issue: 1, page 104-111
• ISSN: 0041-0241

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## Abstract

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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.

## How to cite

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Pestana, Dinis. "A note on Pólya's theorem.." Trabajos de Estadística e Investigación Operativa 35.1 (1984): 104-111. <http://eudml.org/doc/40750>.

@article{Pestana1984,
abstract = {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.},
author = {Pestana, Dinis},
journal = {Trabajos de Estadística e Investigación Operativa},
keywords = {Funciones convexas; Variables aleatorias; Conjuntos convexos; extended Pólya functions; convex set; extreme points; Choquet's theorem; integral representation; characteristic function},
language = {eng},
number = {1},
pages = {104-111},
title = {A note on Pólya's theorem.},
url = {http://eudml.org/doc/40750},
volume = {35},
year = {1984},
}

TY - JOUR
AU - Pestana, Dinis
TI - A note on Pólya's theorem.
JO - Trabajos de Estadística e Investigación Operativa
PY - 1984
VL - 35
IS - 1
SP - 104
EP - 111
AB - 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.
LA - eng
KW - Funciones convexas; Variables aleatorias; Conjuntos convexos; extended Pólya functions; convex set; extreme points; Choquet's theorem; integral representation; characteristic function
UR - http://eudml.org/doc/40750
ER -

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