## The Annals of Probability

### Transforming random elements and shifting random fields

Hermann Thorisson

#### 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.

#### Article information

Source
Ann. Probab., Volume 24, Number 4 (1996), 2057-2064.

Dates
First available in Project Euclid: 6 January 2003

https://projecteuclid.org/euclid.aop/1041903217

Digital Object Identifier
doi:10.1214/aop/1041903217

Mathematical Reviews number (MathSciNet)
MR1415240

Zentralblatt MATH identifier
0879.60051

#### Citation

Thorisson, Hermann. Transforming random elements and shifting random fields. Ann. Probab. 24 (1996), no. 4, 2057--2064. doi:10.1214/aop/1041903217. https://projecteuclid.org/euclid.aop/1041903217

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