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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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*.

- I. Csiszár,
*Some problems concerning measures on topological spaces and convolutions of measures on topological groups*, Les probabilités sur les structures algébriques (Actes Colloq. Internat. du CNRS, No. 186, Clermont-Ferrand, 1969) Editions Centre Nat. Recherche Sci., Paris, 1970, pp. 75–96 (English, with French summary). MR**0420768**
---, - I. Csiszár and Balram S. Rajput,
*A convergence of types theorem for probability measures on topological vector spaces with applications to stable laws*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**36**(1976), no. 1, 1–7. MR**420761**, DOI https://doi.org/10.1007/BF00533204 - Alejandro de Acosta,
*Stable measures and seminorms*, Ann. Probability**3**(1975), no. 5, 865–875. MR**391202**, DOI https://doi.org/10.1214/aop/1176996273 - J. Kuelbs and V. Mandrekar,
*Domains of attraction of stable measures on a Hilbert space*, Studia Math.**50**(1974), 149–162. MR**345155**, DOI https://doi.org/10.4064/sm-50-2-149-162 - Balram S. Rajput,
*On the support of certain symmetric stable probability measures on ${\rm TVS}$*, Proc. Amer. Math. Soc.**63**(1977), no. 2, 306–312. MR**445594**, DOI https://doi.org/10.1090/S0002-9939-1977-0445594-2 - Balram S. Rajput and N. N. Vakhania,
*On the support of Gaussian probability measures on locally convex topological vector spaces*, Multivariate analysis, IV (Proc. Fourth Internat. Sympos., Dayton, Ohio, 1975) North-Holland, Amsterdam, 1977, pp. 297–309. MR**0458520** - Helmut H. Schaefer,
*Topological vector spaces*, Springer-Verlag, New York-Berlin, 1971. Third printing corrected; Graduate Texts in Mathematics, Vol. 3. MR**0342978** - A. Tortrat,
*Structure des lois indéfiniment divisibles $(\mu \,\in \,{\cal I}={\cal I}(X))$ dans un espace vectoriel topologique (séparé) $X$*, Symposium on Probability Methods in Analysis (Loutraki, 1966) Springer, Berlin, 1967, pp. 299–328 (French). MR**0226692**

*On the*$wea{k^\ast }$

*convergence of convolution in a convolution algebra over an arbitrary group*, Studia Sci. Math. Hungar.

**6**(1971), 27-40.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
60B05

Retrieve articles in all journals with MSC: 60B05

Additional Information

Keywords:
Locally convex topological vector space,
infinitely divisible and stable probability measures,
Gaussian probability measure,
topological support

Article copyright:
© Copyright 1977
American Mathematical Society