Tail-fields of products of random variables and ergodic equivalence relations
1999 ◽
Vol 19
(5)
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pp. 1325-1341
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Keyword(s):
We prove the following result. Let $G$ be a countable discrete group with finite conjugacy classes, and let $(X_n, n\in\mathbb Z)$ be a two-sided, strictly stationary sequence of $G$-valued random variables. Then $\mathscr T_\infty =\mathscr T_\infty ^*$, where $\mathscr T_\infty$ is the two-sided tail-sigma-field $\bigcap_{M\ge1}\sigma (X_m:|m|\ge M)$ of $(X_n)$ and $T_\infty ^*$ the tail-sigma-field $\bigcap_{M\ge0}\sigma (Y_{m,n}:m,n\ge M)$ of the random variables $(Y_{m,n}, m,n\ge0)$ defined as the products $Y_{m,n}=X_n\dots X_{-m}$. This statement generalises a number of results in the literature concerning tail triviality of two-sided random walks on certain discrete groups.
2018 ◽
Vol 69
(3)
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pp. 1047-1051
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2018 ◽
Vol 33
(10)
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pp. 1850055
1994 ◽
Vol 121
(2)
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pp. 455-485
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1989 ◽
Vol 17
(4)
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pp. 1635-1645
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1987 ◽
Vol 106
◽
pp. 143-162
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2019 ◽
Vol 18
(08)
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pp. 1950155
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