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

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 ${\sigma }_n\ne 0$ 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.

arXiv comment5 pages, supersedes arXiv:1608.03784 "Cameron-Martin theorems for sequences of symmetric Cauchy-distributed random variables"

Cited by in corpus (2)

• Γ-convergence of Onsager-Machlup functionals. Part II: Infinite product measures on Banach spaces
• An equivalence criterion for infinite products of Cauchy measures