Stable positive definite functions
HTML articles powered by AMS MathViewer
- 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
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