## Electronic Communications in Probability

### 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 $\sigma _n\neq 0$ 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
Accepted: 29 January 2018
First available in Project Euclid: 21 February 2018

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

Digital Object Identifier
doi:10.1214/18-ECP113

Mathematical Reviews number (MathSciNet)
MR3771766

Zentralblatt MATH identifier
1390.60149

#### 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

#### References

• [1] V. I. Bogachev, Gaussian measures, Mathematical Surveys and Monographs, vol. 62, American Mathematical Society, Providence, RI, 1998.
• [2] V. I. Bogachev, Differentiable measures and the Malliavin calculus, Mathematical Surveys and Monographs, vol. 164, American Mathematical Society, Providence, RI, 2010.
• [3] I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, eighth ed., Elsevier/Academic Press, Amsterdam, 2015, Translated from the Russian, Translation edited and with a preface by Daniel Zwillinger and Victor Moll, Revised from the seventh edition [MR2360010].
• [4] S. Kakutani, On equivalence of infinite product measures, Ann. of Math. (2) 49 (1948), 214–224.
• [5] L. A. Shepp, Distingunishing a sequence of random variables from a translate of itself, Ann. Math. Statist. 36 (1965), 1107–1112.
• [6] F. G. Tricomi and A. Erdélyi, The asymptotic expansion of a ratio of gamma functions, Pacific J. Math. 1 (1951), 133–142.