Dirichlet curves, convex order and Cauchy distribution

If α is a probability on Rd and t > 0, the Dirichlet random probability Pt - D(tα) is such that for any measurable partition (A0 , Ak) of Rd the random variable (Pt(A0) ,Pt(Ak)) is Dirichlet distributed with parameters (tα(A0), tα(Ak)). If Rd log(1 + x α(dx) < the random variable Rd xPt (dx) of Rd does exist: let μ(tα) be its distribution. The Dirichlet curve associated to the probability α is the map t μ(tα). It has simple properties like limt-0 μ(tα) = α and limtμ(tα) = δm when m = Rd xα(dx) exists. The present paper shows that if m exists and if is a convex function on Rd then t Rd (x)μ(tα)(dx) is a decreasing function, which means that t μ(tα) 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 μ(tα) = μ(sα) for some 0 ≤ s < t, can we claim that μ is Cauchy distributed in Rd ?