Open Access
January, 1986 Absolute Continuity of Stable Seminorms
T. Byczkowski, K. Samotij
Ann. Probab. 14(1): 299-312 (January, 1986). DOI: 10.1214/aop/1176992629


Suppose that $E$ is a complete separable real metric vector space. It is proved that if $X$ is a symmetric $E$-valued $p$-stable random vector, $0 < p < 2$, and $q$ is a lower semicontinuous, a.s. finite seminorm, then the distribution of $q(X)$ is absolutely continuous apart from a possible jump. If, additionally, $q$ is strictly convex or $0 < p < 1$, then the distribution of $q(X)$ is either absolutely continuous or degenerate at 0. This result settles, in particular, the problem of absolute continuity of the supremum of stable sequences, extending thus Tsirel'son's theorem.


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T. Byczkowski. K. Samotij. "Absolute Continuity of Stable Seminorms." Ann. Probab. 14 (1) 299 - 312, January, 1986.


Published: January, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0601.60004
MathSciNet: MR815972
Digital Object Identifier: 10.1214/aop/1176992629

Primary: 60B05
Secondary: 60E07

Keywords: Absolute continuity , semigroups of measures , seminorms , Stable measures

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 1 • January, 1986
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