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

TL;DR

This paper investigates when infinite products of Cauchy measures remain equivalent under non-unitary dilations, revealing that equivalence depends on the square-summability of the scale factors' deviations from one.

Contribution

It establishes a precise criterion based on square-summability for the equivalence of countable products of Cauchy measures under non-unitary dilations.

Findings

Measures are equivalent iff the scale deviations are square-summable.

Uses Kakutani's theorem on infinite product measures.

Provides a condition for measure equivalence under scale transformations.

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 $(\vert \sigma_n\vert-1)_{n\in\mathbb{N}}$ is square-summable.