Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Anderson inequality is strict for Gaussian and stable measures

Authors: Maciej Lewandowski, Michał Ryznar and Tomasz Żak
Journal: Proc. Amer. Math. Soc. 123 (1995), 3875-3880
MSC: Primary 60B11; Secondary 60G15
MathSciNet review: 1264821
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Abstract: Let $ \mu $ be a symmetric Gaussian measure on a separable Banach space $ (E,\left\Vert \bullet \right\Vert)$. Denote $ U = \{ x:\left\Vert x \right\Vert < 1\} $. Then for every $ x \in {\text{supp}}\;\mu ,x \ne 0$, the function $ t \to \mu (U + tx)$ is strictly decreasing for $ t \in (0,\infty )$. The same property holds for symmetric p-stable measures on E. Using this property we answer a question of W. Linde : if $ \smallint_{U + z} {xd\mu (x) = 0} $, then $ z = 0$.

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Keywords: Gaussian measures, stable measures, Anderson inequality
Article copyright: © Copyright 1995 American Mathematical Society