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Characterizations of invariant distributions

Published online by Cambridge University Press:  24 October 2008

Timothy C. Brown ,
Donald I. Cartwright  and
G. K. Eagleson
Timothy C. Brown
Affiliation:
Department of Statistics, University of Melbourne, Parkville, Vic., 3052
Donald I. Cartwright
Affiliation:
Department of Pure Mathematics, University of Sydney, N.S.W., 2006
G. K. Eagleson
Affiliation:
C.S.I.R.O., Division of Mathematics and Statistics, P.O. Box 218, Lindfield, N.S.W., 2070
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Abstract

Let (S, ρ) be a separable metric space and G a group of transformations of S. Necessary and sufficient conditions for a distribution on S to be invariant under G are derived in terms of the behaviour of the convolution of a random transformation from G and a random element of S.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 2 , March 1985 , pp. 349 - 355
Copyright
Copyright © Cambridge Philosophical Society 1985

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References

REFERENCES

[1]Billingsley, P.. Convergence of Probability Measures. (Wiley, 1968).Google Scholar
[2]Driscoll, M. F.. On pairwise and mutual independence: characterizations of rectangular distributions. J. Amer. Statist. Assoc. 73 (1978), 432–433.Google Scholar
[3]Fisz, M.. Probability Theory and Mathematical Statistics. (Wiley, 1963).Google Scholar
[4]Heyer, H.. Probability Measures on Locally Compact Groups. (Springer-Verlag. 1977).CrossRefGoogle Scholar
[5]Lévy, P.. L'addition des variables aléatoires définies sur une circonférence. Bull. Soc. Math. France, 67 (1939), 1–41.Google Scholar
[6]Revuz, D.. Markov Chains. (North-Holland, 1975).Google Scholar
[7]Rosenblatt, M.. Markov Processes. Structure and Asymptotic Behaviour. (Springer-Verlag, 1971).Google Scholar
[8]Segall, I. E. and Kunze, R. A.. Integrals and Operators. (McGraw-Hill, 1978).CrossRefGoogle Scholar
[9]Thornett, M. L.. The role of randomization in model-based inference. Austral. J. Statist. 24 (1982), 137–145.Google Scholar