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

Electronic Communications in Probability

Han Cheng Lie¹ 

关键词: Cauchy distribution;    change of measure;    equivalence of measure;    random sequence;   

DOI  :  10.1214/18-ECP113

学科分类：统计和概率

来源: Institute of Mathematical Statistics

【 摘 要 】

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 $(\left \vert{\sigma _n} \right \vert -1)_{n\in \mathbb{N} }$ is square-summable.

【 授权许可】

CC BY