The Markov–Krein correspondence in several dimensions

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
$$ \biggl\langle\frac1{1-z_1F_1(M)-\ldots-z_mF_m(M)}\biggr\rangle=\exp\int\ln\frac1{1-z_1f_1(x)-\ldots-z_mf_m(x)}\tau(dx), $$
where $F_k(M)=\int_Xf_k(x)M(dx)$, the angle brackets denote the average in $M$, and $f_1,\dots,f_m$ are the coordinates of a map $f\colon X\to\mathbb R^m$. The formula describes implicitly the joint distribution of the random variables $F_k(M), k=1,\ldots,m$. Assuming that the joint moments $p_{k_1,\dots,k_m}=\int f^{k_1}_1\dots f^{k_m}_m(x)\,d\tau(x)$ are all finite, we restate the above formula as an explicit description of the joint moments of the variables $F_1,\dots,F_m$ in terms of $p_{k_1,\dots,k_m}$. In the case of a finite space, $|X|=N+1$, the problem is to describe the image $\mu$ of a Dirichlet distribution
$$ \frac{M^{\tau_0-1}_0M^{\tau_1-1}_1\dots M^{\tau_N-1}_N}{\Gamma(\tau_0)\Gamma(\tau_1)\dots\Gamma(\tau_N)}dM_1\dots dM_N; \qquad M_0,\dots,M_N\ge, M_0+\ldots+M_N=1 $$
on the $N$-dimensional simplex $\Delta^N$ under a linear map $f\colon\Delta^N\to\mathbb R^m$. An explicit formula for the destiny of $\mu$ was already known in the case of $m=1$; here we find it in the case of $m=N-1$.