## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 23 (2018), paper no. 8, 6 pp.

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

Han Cheng Lie and T.J. Sullivan

#### 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.

#### Article information

**Source**

Electron. Commun. Probab., Volume 23 (2018), paper no. 8, 6 pp.

**Dates**

Received: 30 November 2016

Accepted: 29 January 2018

First available in Project Euclid: 21 February 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1519182083

**Digital Object Identifier**

doi:10.1214/18-ECP113

**Mathematical Reviews number (MathSciNet)**

MR3771766

**Zentralblatt MATH identifier**

1390.60149

**Subjects**

Primary: 60G30: Continuity and singularity of induced measures

Secondary: 60G20: Generalized stochastic processes 60E07: Infinitely divisible distributions; stable distributions

**Keywords**

Cauchy distribution change of measure equivalence of measure random sequence

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Lie, Han Cheng; Sullivan, T.J. Quasi-invariance of countable products of Cauchy measures under non-unitary dilations. Electron. Commun. Probab. 23 (2018), paper no. 8, 6 pp. doi:10.1214/18-ECP113. https://projecteuclid.org/euclid.ecp/1519182083