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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Author: Tilmann Gneiting
Journal: Proc. Amer. Math. Soc. 128 (2000), 1721-1728
MSC (1991): Primary 42A82, 60E10; Secondary 42A24, 42A38
Published electronically: October 27, 1999
MathSciNet review: 1646306
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Abstract: Let $\varphi : [0,\infty) \to \mathbb{R}$ be a continuous function with $\varphi(0) = 1$ and $\lim _{t \to \infty} \varphi(t)$ $= 0$. 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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Additional Information

Tilmann Gneiting
Affiliation: Department of Statistics, Box 354322, University of Washington, Seattle, Washington 98195
Email: tilmann@stat.washington.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-99-05200-4
PII: S 0002-9939(99)05200-4
Received by editor(s): July 13, 1998
Published electronically: October 27, 1999
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society