N. Tsilevich, A. Vershik, M. Yor
Quasi-invariance of the gamma process and multiplicative properties of the Poisson--Dirichlet measures
Preprint series: St. Petersburg Mathematical Society Preprints

MSC:

60E07 Infinitely divisible distributions; stable distributions

60G20 Generalized stochastic processes

Abstract: In this paper we describe new fundamental properties of the law $P_\Gamma$ of
the classical gamma process and related properties of the Poisson--Dirichlet
measures $PD(\theta)$. We prove the quasi-invariance of the measure $P_\Gamma$
with respect to an infinite-dimensional multiplicative group and the
Markov--Krein identity as corollaries of the formula for the Laplace transform
of $P_\Gamma$.
The quasi-invariance of the measure $P_\Gamma$ allows us to obtain new
quasi-invariance properties of the measure $PD(\theta)$. The corresponding
invariance properties hold for $\sigma$-finite analogues of $P_\Gamma$ and
$PD(\theta)$. We also show that the measure $P_\Gamma$ can be considered as a
limit of measures corresponding to the $\alpha$-stable Levy processes when
parameter $\alpha$ tends to zero.
Our approach is based on simultaneous considering the gamma process (especially
its Laplace transform) and its simplicial part -- the Poisson--Dirichlet
measures.

Notes: AMS-TeX, 8 pp.
Submitted to C.R.Acad.Sci.Paris