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INICIO |
27 de julio de 2024
A note on Pólya's theorem.
| Título inglés | A note on Pólya's theorem. |
|---|---|
| Título español | Una nota sobre el teorema de Polya. |
| Autor/es | Pestana, Dinis |
| Organización | Dep. Estat. Cent. Estat. Apl. Univ. Lisboa, Lisboa, Portugal |
| Revista | 0041-0241 |
| Publicación | 1984, 35 (1): 104-111, 7 Ref. |
| Tipo de documento | articulo |
| Idioma | Inglés |
| Resumen inglés | The class of extended Pólya functions Ω = {φ: φ is a continuous real valued real function, φ(-t) = φ(t) ≤ φ(0) Î [0,1], límt→∞ φ(t) = c Î [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 Î 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 Î 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. |
| Clasificación UNESCO | 120206 |
| Palabras clave español | Funciones convexas ; Variables aleatorias ; Conjuntos convexos |
| Código MathReviews | MR0829915 |
| Código Z-Math | Zbl 0733.60034 |
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