Support and seminorm integrability theorems for r-semistable probability measures on LCTVS
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.
KeywordsHilbert Space Probability Measure Convex Cone Topological Vector Space Stable Measure
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