The Dirichlet curve of a probability in ℝ^d

2014/05/19 by Gérard Letac, Gerard Letac, Letac, Gerard +2
Mathematics · #60G57 #62E10 #Advanced Harmonic Analysis Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Probability (math.PR) #advanced mathematical theories #math.PR #msc:60G57 #msc:62E10

paper · pdf · doi:10.48550/arxiv.1405.4744

arxiv created 2014/05/19 · openalex publication_date 2014/05/19 · arxiv updated 2014/05/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

If α is a probability on ℝ^d and t>0, consider the Dirichlet random probability P_t\simD(tα) ; it is such that for any measurable partition (A₀,…,A_k) of ℝ^d then (P_t(A₀),…,P_t(A_k)) is Dirichlet distributed with parameters (tα(A₀)…,tα(A_k)). If ∫_ℝ^dlog(1+‖x‖)α(dx)<∞ the random variable ∫_ℝ^dxP_t(dx) of ℝ^d does exist and we denote by μ(tα) its distribution. The Dirichlet curve associated to the probability α is the map t↦ μ(tα). It has simple properties like lim_{t\searrow 0}μ(tα)=α and lim_{t→ ∞}μ(tα)=δ_m when m=∫_ℝ^d xα(dx) exists. The present paper shows first that if m exists and if ψ is a convex function on ℝ^d then t↦ ∫_ℝ^dψ(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 ℝ^d?

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