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Linear functionals and Markov chains associated with Dirichlet processes

Published online by Cambridge University Press:  04 October 2011

Paul D. Feigin  and
Richard L. Tweedie
Paul D. Feigin
Affiliation:
Technion, Haifa and Division of Mathematics and Statistics, CSIRO
Richard L. Tweedie
Affiliation:
Siromath Pty. Ltd., Sydney and Bond University, Gold Coast, Australia
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Abstract

By investigating a Markov chain whose limiting distribution corresponds to that of the Dirichlet process we are able directly to ascertain conditions for the existence of linear functionals of that process. Together with earlier analyses we are able to characterize those functionals which are a.s. finite in terms of the parameter measure of the process. We also show that the appropriate Markov chain in the space of measures is only weakly convergent and not Harris ergodic.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 105 , Issue 3 , May 1989 , pp. 579 - 585
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

[1] Blackwell, D.. Discreteness of Ferguson selections. Ann. Statist. 1 (1973), 356–358.CrossRefGoogle Scholar
[2] Blackwell, D. and MacQueen, J. B.. Ferguson distributions via Polya urn schemes. Ann. Statist. 1 (1973), 353–355.CrossRefGoogle Scholar
[3] Doss, H. and Sellke, T.. The tails of probabilities chosen from a Dirichlet prior. Ann. Statist. 10 (1982), 1302–1305.CrossRefGoogle Scholar
[4] Ferguson, T. S.. A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 (1973), 209–230.CrossRefGoogle Scholar
[5] Ferguson, T. S.. Prior distributions on spaces of probability measures. Ann. Statist. 2 (1974), 615–629.CrossRefGoogle Scholar
[6] Hannum, R. C., Hollander, M. and Langberg, N. A.. Distributional results for random functionals of a Dirichlet process. Ann. Probab. 9 (1981), 665–670.CrossRefGoogle Scholar
[7] Kallenberg, O.. Random Measures (Academic Press, 1976).Google Scholar
[8] Rosenblatt, M.. Markov Processes: Structure and Asymptotic Behavior (Springer-Verlag, 1971).CrossRefGoogle Scholar
[9] Sethuranam, J. and Tiwari, R.. Convergence of Dirichlet measures and the interpretation of their parameter. In Proceeding of the Third Purdue Symposium on Statistical Decision Theory Related Topics (Academic Press, 1982).Google Scholar
[10] Tweedie, R. L.. Criteria for classifying general Markov chains. Adv. in Appl. Probab. 8 (1976), 737–771.CrossRefGoogle Scholar
[11] Tweedie, R. L.. Topological aspects of Doeblin decompositions for Markov chains. Z. Warsch. Verw. Gebiete 46 (1979), 299–305.CrossRefGoogle Scholar
[12] Tweedie, R. L.. Criteria for rates of convergence of Markov chains, with application to queueing and storage theory. In Probability, Statistics and Analysis (Cambridge University Press, 1983).Google Scholar
[13] Yamato, H.. Characteristic functions of means of distributions chosen from a Dirichlet process. Ann. Probab. 12 (1984), 262–267.CrossRefGoogle Scholar