Riesz exponential families on homogeneous cones

Abstract

In this paper, we introduce, for a multiplier $\chi$, a notion of generalized power function $x\mapsto {\Delta }_{\chi }(x),$ defined on the homogeneous cone $\mathcal{P}$ of a Vinberg algebra $\mathcal{A}$. We then extend to $\mathcal{A}$ the famous Gindikin result, that is we determine the set of multipliers $\chi$ such that the map $\theta \mapsto {\Delta }_{\chi }({\theta }^{-1})$, defined on ${\mathcal{P}}^{\ast }$, is the Laplace transform of a positive measure $R_{\chi }$. We also determine the set of $\chi$ such that $R_{\chi }$ generates an exponential family, and we calculate the variance function of this family