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On the support of symmetric infinitely divisible and stable probability measures on LCTVS

Author: Balram S. Rajput
Journal: Proc. Amer. Math. Soc. 66 (1977), 331-334
MSC: Primary 60B05
MathSciNet review: 0494351
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Abstract: It is shown that the topological support (supp.) of a $ \tau $-regular, symmetric, infinitely divisible (resp. stable of any index $ \alpha \in (0,2)$) probability measure on a Hausdorff LCTVS E is a subgroup (resp. a subspace) of E. The part regarding the support of a stable probability measure of this theorem completes a result of A. De-Acosta [Ann. of Probability 3 (1975), 865-875], who proved a similar result for $ \alpha \in (1,2)$, and the author [Proc. Amer. Math. Soc. 63 (1977), 306-312], who proved it for $ \alpha \in [1,2)$. Further, it provides a complete affirmative solution to the question, raised by J. Kuelbs and V. Mandrekar [Studia Math. 50 (1974), 149-162], of whether the supp. of a symmetric stable probability measure of index $ \alpha \in (0,1]$ on a separable Hilbert space H is a subspace of H.

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Keywords: Locally convex topological vector space, infinitely divisible and stable probability measures, Gaussian probability measure, topological support
Article copyright: © Copyright 1977 American Mathematical Society

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