Electronic Communications in Probability

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

Han Cheng Lie and T.J. Sullivan

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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

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


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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.