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Teor. Veroyatnost. i Primenen., 2001, Volume 46, Issue 4, Pages 697–712 (Mi tvp3795)  

This article is cited in 6 scientific papers (total in 6 papers)

Poisson Measures Quasi-Invariant with Respect to Multiplicative Transformations

M. A. Lifshits, E. Yu. Shmileva

St. Petersburg State University, Department of Mathematics and Mechanics

Abstract: In this work the necessary and sufficient conditions are given for the quasi-invariance of the distributions of Poisson measures on $X\times\mathbf{R}^+$ (for arbitrary measurable space $X$) with respect to a large group of the scalings of the component $\mathbf{R}^+$. It is shown that the class of quasi-invariant measures is far from being exhausted by the measures absolutely continuous with respect to the gamma measure considered in [N. Tsilevich and A. Vershik, C. R. Acad. Sci. Paris Ser. I Math., 329 (1999), pp. 163–168] and [N. Tsilevich, A. Vershik, and M. Yor, Prepublication 575, Universites Paris VI & Paris VII, Paris, 2000]. A criterion is given for the absolute continuity of a Poisson measure with respect to another Poisson measure on an arbitrary measurable space.

Keywords: Poisson measure, spectral measure, quasi-invariance, gamma measure, Hellinger–Kakutani distance.

DOI: https://doi.org/10.4213/tvp3795

Full text: PDF file (1474 kB)

English version:
Theory of Probability and its Applications, 2002, 46:4, 652–666

Bibliographic databases:

Received: 01.03.2001

Citation: M. A. Lifshits, E. Yu. Shmileva, “Poisson Measures Quasi-Invariant with Respect to Multiplicative Transformations”, Teor. Veroyatnost. i Primenen., 46:4 (2001), 697–712; Theory Probab. Appl., 46:4 (2002), 652–666

Citation in format AMSBIB
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\paper Poisson Measures Quasi-Invariant with Respect to Multiplicative Transformations
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\pages 652--666
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. E. Yu. Shmileva, “Small ball probability for centered Poisson process of high intensity”, J. Math. Sci. (N. Y.), 128:1 (2005), 2656–2668  mathnet  crossref  mathscinet  zmath
    2. Nourdin I., Simon T., “On the absolute continuity of Levy processes with drift”, Annals of Probability, 34:3 (2006), 1035–1051  crossref  mathscinet  zmath  isi  scopus
    3. S. S. Gribkova, “The measure preserving transformations of the jump Lévy process”, J. Math. Sci. (N. Y.), 167:4 (2010), 506–511  mathnet  crossref  elib
    4. Kondratiev Yu., Lytvynov E., Vershik A., “Laplace Operators on the Cone of Radon Measures”, J. Funct. Anal., 269:9 (2015), 2947–2976  crossref  mathscinet  zmath  isi  elib  scopus
    5. Hagedorn D., Kondratiev Yu., Lytvynov E., Vershik A., “Laplace Operators in Gamma Analysis”, Stochastic and Infinite Dimensional Analysis, Trends in Mathematics, eds. Bernido C., CarpioBernido M., Grothaus M., Kuna T., Oliveira M., DaSilva J., Birkhauser Boston, 2016, 119–147  crossref  mathscinet  isi
    6. Ibraheem H.O., Lytvynov E., “Quasi-Invariance of Completely Random Measures”, Methods Funct. Anal. Topol., 24:3 (2018), 207–239  mathscinet  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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