Dirichlet curves, convex order and Cauchy distribution

Abstract

If $\alpha$ is a probability on $\mathbb{R}^{d}$ and $t>0$, the Dirichlet random probability $P_{t}\sim\mathcal{D}(t\alpha)$ is such that for any measurable partition $(A_{0},\ldots,A_{k})$ of $\mathbb{R}^{d}$ the random variable $(P_{t}(A_{0}),\ldots,P_{t}(A_{k}))$ is Dirichlet distributed with parameters $(t\alpha(A_{0}),\ldots,t\alpha(A_{k}))$. If $\int_{\mathbb{R}^{d}}\log(1+\Vert x\Vert )\alpha(dx)<\infty$ the random variable $\int_{\mathbb{R}^{d}}xP_{t}(dx)$ of $\mathbb{R}^{d}$ does exist: let $\mu(t\alpha)$ be its distribution. The Dirichlet curve associated to the probability $\alpha$ is the map $t\mapsto\mu(t\alpha)$. It has simple properties like $\lim_{t\searrow0}\mu(t\alpha)=\alpha$ and $\lim_{t\rightarrow\infty}\mu(t\alpha)=\delta_{m}$ when $m=\int_{\mathbb{R}^{d}}x\alpha(dx)$ exists. The present paper shows that if $m$ exists and if $\psi$ is a convex function on $\mathbb{R}^{d}$ then $t\mapsto\int_{\mathbb{R}^{d}}\psi(x)\mu(t\alpha)(dx)$ is a decreasing function, which means that $t\mapsto\mu(t\alpha)$ is decreasing according to the Strassen convex order of probabilities. The second aim of the paper is to prove a group of results around the following question: if $\mu(t\alpha)=\mu(s\alpha)$ for some $0\leq s<t$, can we claim that $\mu$ is Cauchy distributed in $\mathbb{R}^{d}?$