### Dilations of Cauchy measures in Electron. Commun. Prob.

“Quasi-invariance of countable products of Cauchy measures under non-unitary dilations”, by Han Cheng Lie and myself, has just appeared online in *Electronic Communications in Probability*.
This main result of this article can be understood as an analogue of the celebrated Cameron–Martin theorem, which characterises the directions in which an infinite-dimensional Gaussian measure can be translated while preserving equivalence of the original and translated measure;
our result is a similar characterisation of equivalence of measures, but for infinite-dimensional Cauchy measures under dilations instead of translations.

H. C. Lie & T. J. Sullivan. “Quasi-invariance of countable products of Cauchy measures under non-unitary dilations.” *Electronic Communications in Probability* **23**(8):1–6, 2018. doi:10.1214/18-ECP113

**Abstract.** Consider an infinite sequence \( (U_{n})_{n \in \mathbb{N}} \) of independent Cauchy random variables, defined by a sequence \( (\delta_{n})_{n \in \mathbb{N}} \) of location parameters and a sequence \( (\gamma_{n})_{n \in \mathbb{N}} \) of scale parameters. Let \( (W_{n})_{n \in \mathbb{N}} \) be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence \( (\sigma_{n} \gamma_{n})_{n \in \mathbb{N}} \) of scale parameters, with \( \sigma_{n} \neq 0 \) for all \( n \in \mathbb{N} \). Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of \( (U_{n})_{n \in \mathbb{N}} \) and \( (W_{n})_{n \in \mathbb{N}} \) are equivalent if and only if the sequence \( (| \sigma_{n}| - 1 )_{n \in \mathbb{N}} \) is square-summable.

Published on Wednesday 21 February 2018 at 09:30 UTC #publication #electron-commun-prob #cauchy-distribution