The Dirichlet curve of a probability in $\mathbb{R}^d$

TL;DR

This paper studies the properties of the Dirichlet curve associated with a probability measure in , showing its monotonicity in convex order and exploring conditions under which the curve's equality implies Cauchy distribution.

Contribution

It introduces the Dirichlet curve in , proves its monotonicity in convex order, and investigates when equality of the curve at different points implies a Cauchy distribution.

Findings

The Dirichlet curve  is decreasing in convex order.

The curve converges to the original measure as t approaches 0.

Equality of the curve at different parameters implies the measure is Cauchy.

Abstract

If $\alpha$ is a probability on $\mathbb{R}^d$ and $t>0,$ consider the Dirichlet random probability $P_t\sim\mathcal{D}(t\alpha) ;$ it is such that for any measurable partition $(A_0,\ldots,A_k)$ of $\mathbb{R}^d$ then $(P_t(A_0),\ldots,P_t(A_k))$ is Dirichlet distributed with parameters $(t\alpha(A_0)\ldots,t\alpha(A_k)).$ If $\int_{\mathbb{R}^d}\log(1+\|x\|)\alpha(dx)<\infty$ the random variable $\int_{\mathbb{R}^d}xP_t(dx)$ of $\mathbb{R}^d$ does exist and we denote by $\mu(t\alpha)$ its distribution. The Dirichlet curve associated to the probability $\alpha$ is the map $t\mapsto \mu(t\alpha).$ It has simple properties like $\lim_{t\searrow 0}\mu(t\alpha)=\alpha$ and $\lim_{t\rightarrow \infty}\mu(t\alpha)=\delta_m$ when $m=\int_{\mathbb{R}^d} x\alpha(dx)$ exists. The present paper shows first 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 $\mathbb{R}^d?$