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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
DOI: https://doi.org/10.1090/S0002-9939-1977-0494351-X
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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  • [1] I. Csiszár, Some problems concerning measures on topological spaces and convolutions of measures on topological groups, Les Probabilités sur les Structures Algébriques (Colloques Internat. de CNRS, Paris, 1970), Clermont-Ferrand, pp. 75-79. MR 0420768 (54:8780)
  • [2] -, On the $ wea{k^\ast}$ convergence of convolution in a convolution algebra over an arbitrary group, Studia Sci. Math. Hungar. 6 (1971), 27-40.
  • [3] I. Csiszár and B. S. Rajput, A convergence of types theorem for probability measures on TVS with applications to stable laws, Z. Wahrscheinlichkeitstheorie und verw. Gebiete 36 (1976), 1-7. MR 0420761 (54:8773)
  • [4] A. De-Acosta, Stable measures and seminorms, Ann. of Probability 3 (1975), 865-875. MR 0391202 (52:12023)
  • [5] J. Kuelbs and V. Mandrekar, Domains of attractions of stable measures on Hilbert space, Studia. Math. 50 (1974), 149-162. MR 0345155 (49:9894)
  • [6] B. S. Rajput, On the support of certain symmetric stable probability measures on TVS, Proc. Amer. Math. Soc. 63 (1977), 306-312. MR 0445594 (56:3931)
  • [7] B. S. Rajput and N. N. Vakhania, On the support of Gaussian probability measures on LCTVS, Fourth Internat. Sympos. on Multivariable Analysis (P. R. Krishnaiah, editor), North-Holland, New York, 1977, pp. 297-309. MR 0458520 (56:16720)
  • [8] H. H. Schaefer, Topological vector spaces, 3rd printing, Springer, New York, 1970. MR 0342978 (49:7722)
  • [9] A. Tortrat, Structure des lois indéfiniment divisibles dans un espace vectoriel topologique, Lecture Notes in Math., vol. 31, Springer-Verlag, New York, 1967, pp. 299-327. MR 0226692 (37:2279)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1977-0494351-X
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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