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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Stable positive definite functions
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by K. R. Parthasarathy and K. Schmidt PDF
Trans. Amer. Math. Soc. 203 (1975), 161-174 Request permission

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 $\text {GL}(n,R)$ 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
    P. Lévy, Théorie de l’addition des variables aléatoires, Gauthier-Villars, Paris, 1954.
  • Eugene Lukacs, Characteristic functions, Hafner Publishing Co., New York, 1970. Second edition, revised and enlarged. MR 0346874
  • 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
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 203 (1975), 161-174
  • MSC: Primary 60B15; Secondary 43A35
  • DOI: https://doi.org/10.1090/S0002-9947-1975-0370681-X
  • MathSciNet review: 0370681