  We study a 2-parametric family of probability measures on an
infinite-dimensional simplex (the Thoma simplex). These measures originate in
harmonic analysis on the infinite symmetric group (S.Kerov, G.Olshanski and
A.Vershik, Comptes Rendus Acad. Sci. Paris I 316 (1993), 773-778). Our approach
is to interpret them as probability distributions on a space of point
configurations, i.e., as certain point stochastic processes, and to find the
correlation functions of these processes.
  In the present paper we relate the correlation functions to the solutions of
certain multidimensional moment problems. Then we calculate the first
correlation function which leads to a conclusion about the support of the
initial measures. In the appendix, we discuss a parallel but more elementary
theory related to the well-known Poisson-Dirichlet distribution.
  The higher correlation functions are explicitly calculated in the subsequent
paper (A.Borodin, math.RT/9804087). In the third part (A.Borodin and
G.Olshanski, math.RT/9804088) we discuss some applications and relationships
with the random matrix theory.
  The goal of our work is to understand new phenomena in noncommutative
harmonic analysis which arise when the irreducible representations depend on
countably many continuous parameters.
