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Petersburg Department of Steklov Institute of Mathematics

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PREPRINT 13/1998

N. Tsilevich
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STATIONARY MEASURES ON THE SPACE OF VIRTUAL
PERMUTATIONS FOR AN ACTION OF THE INFINITE
SYMMETRIC GROUP

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~$1$.

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