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Petersburg Department of Steklov Institute of Mathematics

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PREPRINT 1/1998

Sergei Kerov and Natalia Tsilevich
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THE MARKOV-KREIN CORRESPONDENCE
IN SEVERAL DIMENSIONS

This preprint was accepted January, 1998.

Contact:
` S.V.Kerov and N. Tsilevich `

Abstract:

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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