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2018 Quasi-invariance of countable products of Cauchy measures under non-unitary dilations
Han Cheng Lie, T.J. Sullivan
Electron. Commun. Probab. 23: 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 $(\left \vert{\sigma _n} \right \vert -1)_{n\in \mathbb{N} }$ is square-summable.

Citation

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Han Cheng Lie. T.J. Sullivan. "Quasi-invariance of countable products of Cauchy measures under non-unitary dilations." Electron. Commun. Probab. 23 1 - 6, 2018. https://doi.org/10.1214/18-ECP113

Information

Received: 30 November 2016; Accepted: 29 January 2018; Published: 2018
First available in Project Euclid: 21 February 2018

zbMATH: 1390.60149
MathSciNet: MR3771766
Digital Object Identifier: 10.1214/18-ECP113

Subjects:
Primary: 60G30
Secondary: 60E07 , 60G20

Keywords: Cauchy distribution , Change of measure , equivalence of measure , random sequence

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