  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.
