Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-25T02:58:39.391Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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