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

Rajput, Balram S.

doi:10.1090/S0002-9939-1977-0494351-X

On the support of symmetric infinitely divisible and stable probability measures on LCTVS
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Proc. Amer. Math. Soc. 66 (1977), 331-334 Request permission

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.

References

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On the

convergence of convolution in a convolution algebra over an arbitrary group

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