Abstract
Consider a locally compact second countable topological transformation group acting measurably on an arbitrary space. We show that the distributions of two random elements X and $X'$ in this space agree on invariant sets if and only if there is a random transformation $\Gamma$ such that $\Gamma X$ has the same distribution as $X'$. Applying this to random fields in d dimensions under site shifts, we show further that these equivalent claims are also equivalent to site-average total variation convergence. This convergence result extends to amenable groups.
Citation
Hermann Thorisson. "Transforming random elements and shifting random fields." Ann. Probab. 24 (4) 2057 - 2064, October 1996. https://doi.org/10.1214/aop/1041903217
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