Kuttner’s problem and a Pólya type criterion for characteristic functions
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- by Tilmann Gneiting PDF
- Proc. Amer. Math. Soc. 128 (2000), 1721-1728 Request permission
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.References
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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
- Received by editor(s): July 13, 1998
- Published electronically: October 27, 1999
- Communicated by: Christopher D. Sogge
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 1721-1728
- MSC (1991): Primary 42A82, 60E10; Secondary 42A24, 42A38
- DOI: https://doi.org/10.1090/S0002-9939-99-05200-4
- MathSciNet review: 1646306