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