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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Stable positive definite functions


Authors: K. R. Parthasarathy and K. Schmidt
Journal: Trans. Amer. Math. Soc. 203 (1975), 161-174
MSC: Primary 60B15; Secondary 43A35
MathSciNet review: 0370681
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Abstract: This paper investigates the stability of positive definite functions on locally compact groups under one parameter groups of automorphisms. As an application of this it is shown that the only probability distributions on $ {R^n}$ which are stable under the full automorphism group GL$ (n,R)$ of $ {R^n}$ are the nondegenerate Gaussian distributions. It is furthermore shown that there are no nondegenerate probability distributions of $ {R^n}$ which are stable under $ {\text{SL}}(n,R)$.


References [Enhancements On Off] (What's this?)

  • [1] P. Lévy, Théorie de l'addition des variables aléatoires, Gauthier-Villars, Paris, 1954.
  • [2] Eugene Lukacs, Characteristic functions, Hafner Publishing Co., New York, 1970. Second edition, revised and enlarged. MR 0346874 (49 #11595)
  • [3] K. R. Parthasarathy and K. Schmidt, Positive definite kernels, continuous tensor products, and central limit theorems of probability theory, Lecture Notes in Mathematics, Vol. 272, Springer-Verlag, Berlin-New York, 1972. MR 0622034 (58 #29849)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1975-0370681-X
PII: S 0002-9947(1975)0370681-X
Article copyright: © Copyright 1975 American Mathematical Society