N.Tsilevich and A.Vershik.
Quasi-invariance of the gamma process and multiplicative properties of the Poisson-Dirichlet measures.
C. R. Acad. Sci. Paris, 329, Serie I (1999), 163-168.

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 (the fact first discovered in [4]) 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.

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