This preprint was accepted April, 1998
Contact: N. Tsilevich
ABSTRACT: We describe stationary central measures on the space of virtual permutations (that is a projective limit of finite symmetric groups) for an action of the infinite symmetric group. The most important class of central distributions consists of measures $\mu$ such that the sum of normalized cycle lengths is equal to~$1$ for almost all with respect to $\mu$ virtual permutations. In this class, the only stationary distribution is the Ewens measure with parameter~$1$, that is the projective limit of the Haar measures on symmetric groups, and this distribution is invariant. Equivalent setting of the problem is to describe invariant measures for a family of Markovian operators on the simplex of infinite monotone sequences with sum at most~$1$. The ergodic invariant measures are homothetic images of the famous Poisson--Dirichlet distribution PD(1) with parameter~$1$. In particular, we obtain a new characterization of PD(1) as the only invariant distribution on the simplex of sequences with sum~$1$.