S.V.Kerov and N.V.Tsilevich.
The Markov-Krein correspondence in several dimensions.
Zapiski Nauchn. Semin. POMI, 283 (2001), 98-122.
English translation: J. Math. Sci. (N.Y.) 121, No. 3 (2004), 2345-235.

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 $\frac{{M_0^{\tau _{0^{ - 1} } } M_1^{\tau _{1^{ - 1} } } ... M_N^{\tau _{N^{ - 1} } } }}{{\Gamma (\tau _0 )\Gamma (\tau _1 )... \Gamma (\tau _N )}}dM_1 ...dM_{N}$, $M_0 ,...,M_N \geqslant 0$, $M_0 + ... + M_N = 1$, 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.

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