On the Support of Symmetric Infinitely Divisible and Stable Probability Measures on LCTVS

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.