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Distributional properties of means of random probability measures

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Author(s): Antonio Lijoi | Igor Prünster

Journal: Statistics Surveys
ISSN 1935-7516

Volume: 3;
Start page: 47;
Date: 2009;
Original page

Keywords: Bayesian Nonparametrics | Completely random measures | Cifarelli–Regazzini identity | Dirichlet process | Functionals of random probability measures | Generalized Stieltjes transform | Neutral to the right processes | Normalized random measures | Posterior distribution | Random means | Random probability measure | Two–parameter Poisson–Dirichlet process

ABSTRACT
The present paper provides a review of the results concerning distributional properties of means of random probability measures. Our interest in this topic has originated from inferential problems in Bayesian Nonparametrics. Nonetheless, it is worth noting that these random quantities play an important role in seemingly unrelated areas of research. In fact, there is a wealth of contributions both in the statistics and in the probability literature that we try to summarize in a unified framework. Particular attention is devoted to means of the Dirichlet process given the relevance of the Dirichlet process in Bayesian Nonparametrics. We then present a number of recent contributions concerning means of more general random probability measures and highlight connections with the moment problem, combinatorics, special functions, excursions of stochastic processes and statistical physics.

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