Tim Sullivan

Quasi-invariance of countable products of Cauchy measures under non-unitary dilations

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