Canonical correlations for dependent gamma processes

TL;DR

This paper characterizes exchangeable pairs of gamma random measures with dependence described by canonical correlations, linking their finite-dimensional distributions to Dirichlet processes and identifying conditions for independence and process structures.

Contribution

It introduces a novel characterization of dependent gamma processes using canonical correlations and connects them to Dirichlet processes and Dawson--Watanabe diffusions.

Findings

Canonical correlation sequences are moments of Dirichlet process means.

Conditions for gamma measures to have independent joint increments.

Characterization of Dawson--Watanabe diffusions with gamma reversible measures.

Abstract

The present paper provides a characterisation of exchangeable pairs of random measures $(\widetilde\mu_1,\widetilde\mu_2)$ whose identical margins are fixed to coincide 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 $(\widetilde\mu_1,\widetilde\mu_2)$ are moments of means of a Dirichlet process having random base measure. 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 characterised as Dawson--Watanabe diffusions with independent homogeneous immigration, time-changed via an independent subordinator. It is thus shown that Dawson--Watanabe diffusions subordinated by pure drift are the only processes in this class whose time-finite-dimensional distributions have, jointly, independent increments.