Cameron-Martin theorems for sequences of symmetric Cauchy-distributed random variables

TL;DR

This paper investigates conditions under which sequences of symmetric Cauchy-distributed random variables, perturbed by changes in location or scale parameters, have equivalent probability laws, utilizing Kakutani's measure equivalence theorem.

Contribution

It introduces sufficient conditions for the equivalence of laws of perturbed and original Cauchy sequences using infinite product measure theory.

Findings

Established criteria for measure equivalence of Cauchy sequences

Applied Kakutani's theorem to infinite product measures

Provided theoretical insights into Cauchy distribution perturbations

Abstract

Given a sequence of Cauchy-distributed random variables defined by a sequence of location parameters and a sequence of scale parameters, we consider another sequence of random variables that is obtained by perturbing the location or scale parameter sequences. Using a result of Kakutani on equivalence of infinite product measures, we provide sufficient conditions for the equivalence of laws of the two sequences.