Kuttner's problem and a Pólya type criterion for characteristic functions期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Kuttner's problem and a Pólya type criterion for characteristic functions

Authors:	Tilmann Gneiting

Institution:	Department of Statistics, Box 354322, University of Washington, Seattle, Washington 98195

Abstract:	Let $\varphi : 0,\infty) \to \mathbb{R}$ be a continuous function with $\varphi(0) = 1$ and $\lim _{t \to \infty} \varphi(t)$ . If $t^{-1} (\sqrt{t} \, \varphi'(\sqrt{t}) - \varphi'(\sqrt{t}))$ is convex, then $\psi(t) = \varphi(\|t\|)$ , $t \in \mathbb{R}$ , is the characteristic function of an absolutely continuous probability distribution. The criterion complements Pólya's theorem and applies to characteristic functions with various types of behavior at the origin. In particular, it provides upper bounds on Kuttner's function $k(\lambda)$ , $\lambda \in (0,2)$ , which gives the minimal value of $\kappa$ such that $(1-\|t\|^\lambda)_+^\kappa$ is a characteristic function. Specifically, $k(5/3) \leq 3$ . Furthermore, improved lower bounds on Kuttner's function are obtained from an inequality due to Boas and Kac.

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