Abstract
Let μ be an r-semistable K-regular probability measure of index α ε (0, 2] on a complete locally convex topological vector space E. It is shown that the topological support Sμ of μ is a translated convex cone if α ε (0, 1), and a translated truncated cone if α ε (1, 2]. Further, if α=1 and μ is symmetric, then it is shown that Sμ is a vector subspace of E. These results subsume all earlier known results regarding the support of stable measures. A result regarding the support of infinitely divisible probability measure on E is also obtained. A seminorm integrability theorem is obtained for K-regular r-semistable probability measures μ on E. The result of de Acosta (Ann. of Probability, 3(1975), 865 – 875) and Kanter (Trans. Seventh Prague Conf., (1974), 317 – 323) is included in this theorem as long as the measures are defined on LCTVS and seminorm is continuous.
The research of this author was partially supported by the Office of Naval Research under contract No. N00014-78-C-0468.
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© 1980 Springer-Verlag
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Louie, D., Rajput, B.S. (1980). Support and seminorm integrability theorems for r-semistable probability measures on LCTVS. In: Weron, A. (eds) Probability Theory on Vector Spaces II. Lecture Notes in Mathematics, vol 828. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0097403
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DOI: https://doi.org/10.1007/BFb0097403
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