The Markov–Krein Correspondence in Several Dimensions

Abstract

Given a probability distribution $\tau$ on a space X, let M=M $\tau$denote the random probability measure on X known as Dirichlet random measure with parameter distribution $\tau$. We prove the formula $\left\langle {\frac{1}{{1 - z_1 F_1 (M) - ... - z_m F_m (M)}}} \right\rangle = {\text{exp}}\int {{\text{ln}}} \frac{1}{{1 - z_1 f_1 (x) - ... - z_m f_m (x)}}\tau (dx)$ where $F_k (M) = \int_X {f_k } (x)M(dx)$, the angle brackets denote the average in M, and f ₁,...,f _m are the coordinates of a map $f:X \to \mathbb{R}^m$. The formula describes implicitly the joint distribution of the random variables F _k (M), k=1,...,m. Assuming that the joint moments $p_{k_1 ,...,k_m } = \int {f_1^{k_1 } } (x)...f_m^{k_m } (x)d\tau (x)$ are all finite, we restate the above formula as an explicit description of the joint moments of the variables F ₁,...,F _m in terms of $p_{k_1 ,...,k_m }$. In the case of a finite space, |X|=N+1, the problem is to describe the image $\mu$ of a Dirichlet distribution $\frac{{M_0^{\tau _{0^{ - 1} } } M_1^{\tau _{1^{ - 1} } } ...M_N^{\tau _{N^{ - 1} } } }}{{\Gamma (\tau _0 )\Gamma (\tau _1 )...\Gamma (\tau _N )}}dM_1 ...dM_{N;} {\text{ }}M_0 ,...,M_N \geqslant 0,M_0 + ... + M_N = 1$ on the N-dimensional simplex $\Delta ^N$ under a linear map $f:\Delta ^N \to \mathbb{R}^m$. An explicit formula for the density of $\mu$ was already known in the case of m=1; here we find it in the case of m=N-1. Bibliography: 15 titles.