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

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Lie, Han Cheng and Sullivan, T. J. (2018) Quasi-invariance of countable products of Cauchy measures under non-unitary dilations. Electronic Communications in Probability, 23 . 8. doi:10.1214/18-ECP113

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Official URL: http://dx.doi.org/10.1214/18-ECP113

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Abstract

Consider an infinite sequence (Un)n∈N of independent Cauchy random variables, defined by a sequence (δn)n∈N of location parameters and a sequence (γn)n∈N of scale parameters. Let (Wn)n∈N be another infinite sequence of independent Cauchy random variables defined by the same sequence of location parameters and the sequence (σnγn)n∈N of scale parameters, with σn≠0 for all n∈N. Using a result of Kakutani on equivalence of countably infinite product measures, we show that the laws of (Un)n∈N and (Wn)n∈N are equivalent if and only if the sequence (|σn|−1)n∈N is square-summable.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
T Technology > TA Engineering (General). Civil engineering (General)
Divisions: Faculty of Science > Engineering
Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH): Cauchy problem, Differential equations, Partial, Quasi-metric spaces, Infinite
Journal or Publication Title: Electronic Communications in Probability
Publisher: Institute of Mathematical Statistics
ISSN: 1083-589X
Official Date: 21 February 2018
Dates:
DateEvent
21 February 2018Published
29 January 2018Accepted
Date of first compliant deposit: 15 April 2020
Volume: 23
Article Number: 8
DOI: 10.1214/18-ECP113
Status: Peer Reviewed
Publication Status: Published
Publisher Statement: "The right to place the final version of this article Lie, Han Cheng and Sullivan, T. J. (2018) Quasi-invariance of countable products of Cauchy measures under non-unitary dilations. Electronic Communications in Probability, 23. 8. on their own homepage or in a public digital repository, provided there is a link to the official journal site."
Access rights to Published version: Open Access
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
UNSPECIFIED[DFG] Deutsche Forschungsgemeinschafthttp://dx.doi.org/10.13039/501100001659
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