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Random shifts which preserve measure


Authors: Donald Geman and Joseph Horowitz
Journal: Proc. Amer. Math. Soc. 49 (1975), 143-150
MSC: Primary 28A65; Secondary 60G10
DOI: https://doi.org/10.1090/S0002-9939-1975-0396907-X
MathSciNet review: 0396907
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Abstract: Given a flow $ {\theta _g},g \in G$ a group, over a probability space $ (\Omega ,\mathfrak{F},P)$ and a $ G$-valued random variable $ Z$, we exhibit the Lebesgue decomposition of the measure $ P \circ \theta _Z^{ - 1}$ relative to $ P$, and give necessary and sufficient conditions for equality $ (P \circ \theta _Z^{ - 1} = P)$, absolute continuity $ (P \circ \theta _Z^{ - 1} \ll P)$, and singularity $ (P \circ \theta _Z^{ - 1} \bot P)$ in terms of the Haar measure. The proof rests on the theory of ``Palm measures'' as developed by Mecke and the authors. Specializing the group $ G$, we retrieve some known results for the integers and real line, and compute the Radon-Nikodým derivatives in various cases.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0396907-X
Article copyright: © Copyright 1975 American Mathematical Society

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