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Paper No. 12-12

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D Spano and A Lijoi

Canonical correlations for dependent gamma processes

Abstract: The present paper provides a characterization of exchangeable pairs of random measures (_1; _2) whose identical margins are fixed to co-incide with the distribution of a gamma completely random measure, and whose dependence structure is given in terms of canonical correlations. It is first shown that canonical correlation sequences for the finite-dimensional distributions of (_1; _2) are moments of means of a Dirichlet process having random base measure. A few related illustrations are provided, with some of them being of interest for applications to Bayesian statistics. Necessary and sufficient conditions are further given for canonically correlated gamma completely random measures to have independent joint increments. Finally, time-homogeneous Feller processes with gamma reversible measure and canonical autocorrelations are characterized as subordinated Dawson Watanabe diffusions with independent homogeneous immigration. It is thus shown that such Dawson Watanabe diffusions subordinated by pure drift are the only processes in this class whose time-finite-dimensional distributions have, jointly, independent increments. AMS 2000 subject classi_cations: Primary 60G57, 60G51, 62F15.

Keywords: Canonical correlations, Completely random measures, Laguerre polynomials, Dawson-Watanabe processes, Partial exchange-ability, Random Dirichlet means, Extended Gamma processes, Bayesian nonparametrics.