Petersburg Department of Steklov Institute of Mathematics


Sergei Kerov and Natalia Tsilevich


This preprint was accepted January, 1998.
Contact: S.V.Kerov and N. Tsilevich

We compute the joint moments of several linear functionals with respect to a Dirichlet random measure and some of its generalizations. 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 the parameter distribution $\tau$. We prove the formula $$ \Big\langle \frac{1}{1-z_1F_1(M)-\ldots-z_mF_m(M)} \Big\rangle = \exp \int \ln \frac{1}{1-z_1f_1(x)-\ldots-z_mf_m(x)} \tau(dx), $$ where $F_k(M)=\int_Xf_k(x)M(dx)$, the angle brackets denote the average in $M$, and $f_1,\,\ldots,f_m$ are the coordinates of a map $f:X\to\Bbb{R}^m$. The formula describes implicitly the joint distribution of the random variables $F_k(M)$, $k=1,\,\ldots,m$. Assuming that the joint moments $p_{k_1,\ldots,k_m}=\int f_1^{k_1}(x)\ldots f_m^{k_m}(x)d\tau(x)$ are all finite, we restate the above formula as an explicit description of the joint moments of the variables $F_1,\,\ldots,F_m$ in terms of $p_{k_1,\ldots,k_m}$. In case of a finite space, $|X|=N+1$, the problem is to describe the image $\mu$ of a Dirichlet distribution $$ \frac{M_0^{\tau_0-1}M_1^{\tau_1-1}\ldots M_N^{\tau_N-1}} {\Gamma(\tau_0)\Gamma(\tau_1)\ldots\Gamma(\tau_N)} dM_1 \ldots dM_N; \qquad M_0,\,\ldots,M_N\ge0,\; M_0+\ldots+M_N=1 $$ on the $N$-dimensional simplex $\Delta^N$ under a linear map $f:\Delta^N\to\Bbb{R}^m$. An explicit formula for the density of $\mu$ was already known in case $m=1$; here we find it in case $m=N$.

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