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April, 1981 Asymptotic Properties of Semigroups of Measures on Vector Spaces
T. Byczkowski, T. Zak
Ann. Probab. 9(2): 211-220 (April, 1981). DOI: 10.1214/aop/1176994463


Let $(E, B)$ be a measurable vector space and $q$ be a measurable seminorm on $E$. Suppose that $(\mu_t)_{t > 0}$ is a $q$-continuous convolution semigroup of probability measures on $(E, B)$. It is proved that there exists a right-continuous nonincreasing function $\theta$ such that $\lim_{t \rightarrow 0+} (1/t)\cdot \mu_t\{x: q(x) > s\} = \theta(s)$ for every $s > 0$ at which $\theta$ is continuous. If $\mu_t, t > 0$, are Gaussian, then $\theta \equiv 0$; if there exists a measurable linear functional $f$ such that $f(\cdot)$ is not Gaussian (with respect to $\mu_1$) and $q \geqslant |f|$ then $\theta \not\equiv 0$.


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T. Byczkowski. T. Zak. "Asymptotic Properties of Semigroups of Measures on Vector Spaces." Ann. Probab. 9 (2) 211 - 220, April, 1981.


Published: April, 1981
First available in Project Euclid: 19 April 2007

zbMATH: 0462.60002
MathSciNet: MR606984
Digital Object Identifier: 10.1214/aop/1176994463

Primary: 60B05
Secondary: 28A40

Keywords: Gaussian measures , Semigroup of measures , seminorm , Stable measures

Rights: Copyright © 1981 Institute of Mathematical Statistics

Vol.9 • No. 2 • April, 1981
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