Zapiski Nauchnykh Seminarov POMI
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Zap. Nauchn. Sem. POMI:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


Zapiski Nauchnykh Seminarov POMI, 2001, Volume 283, Pages 98–122 (Mi znsl1525)  

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
English version:
Journal of Mathematical Sciences (New York), 2004, Volume 121, Issue 3, Pages 2345–2359
DOI: https://doi.org/10.1023/B:JOTH.0000024616.50649.89
Bibliographic databases:
UDC: 519.21
Language: English
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
Citation in format AMSBIB
\Bibitem{KerTsi01}
\by S.~V.~Kerov, N.~V.~Tsilevich
\paper The Markov--Krein correspondence in several dimensions
\inbook Representation theory, dynamical systems, combinatorial and algoritmic methods. Part~VI
\serial Zap. Nauchn. Sem. POMI
\yr 2001
\vol 283
\pages 98--122
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl1525}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1879065}
\zmath{https://zbmath.org/?q=an:1147.60303}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2004
\vol 121
\issue 3
\pages 2345--2359
\crossref{https://doi.org/10.1023/B:JOTH.0000024616.50649.89}
Linking options:
  • https://www.mathnet.ru/eng/znsl1525
  • https://www.mathnet.ru/eng/znsl/v283/p98
  • This publication is cited in the following 10 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Записки научных семинаров ПОМИ
    Statistics & downloads:
    Abstract page:196
    Full-text PDF :87
     
      Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2024