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Zapiski Nauchnykh Seminarov POMI, 2001, Volume 283, Pages 98–122
(Mi znsl1525)
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This article is cited in 10 scientific papers (total in 10 papers)
The Markov–Krein correspondence in several dimensions
S. V. Kerov, N. V. Tsilevich St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
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
$$
\biggl\langle\frac1{1-z_1F_1(M)-\ldots-z_mF_m(M)}\biggr\rangle=\exp\int\ln\frac1{1-z_1f_1(x)-\ldots-z_mf_m(x)}\tau(dx),
$$
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 $m=N-1$.
Received: 29.10.2001
Citation:
S. V. Kerov, N. V. Tsilevich, “The Markov–Krein correspondence in several dimensions”, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part VI, Zap. Nauchn. Sem. POMI, 283, POMI, St. Petersburg, 2001, 98–122; J. Math. Sci. (N. Y.), 121:3 (2004), 2345–2359
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https://www.mathnet.ru/eng/znsl1525 https://www.mathnet.ru/eng/znsl/v283/p98
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