Random shifts which preserve measure
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- by Donald Geman and Joseph Horowitz PDF
- Proc. Amer. Math. Soc. 49 (1975), 143-150 Request permission
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.References
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Additional Information
- © Copyright 1975 American Mathematical Society
- 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