Electronic Communications in Probability

A characterization of limiting functions arising in Mod-* convergence

Emmanuel Kowalski, Joseph Najnudel, and Ashkan Nikeghbali

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In this note, we characterize the limiting functions in mod-Gaussian convergence; our approach sheds a new light on the nature of mod-Gaussian convergence as well. Our results in fact more generally apply to  mod-* convergence, where * stands for any family of probability distributions whose Fourier transforms do not vanish. We moreover provide new examples, including two new examples of (restricted) mod-Cauchy convergence from arithmetics related to Dedekind sums and the linking number of modular geodesics.

Article information

Electron. Commun. Probab., Volume 20 (2015), paper no. 79, 11 pp.

Accepted: 31 October 2015
First available in Project Euclid: 7 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60B10: Convergence of probability measures
Secondary: 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60E05: Distributions: general theory 11F20: Dedekind eta function, Dedekind sums

Mod-* convergence Fourier transform limiting functions distributions Dedekind sums modular geodesics

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Kowalski, Emmanuel; Najnudel, Joseph; Nikeghbali, Ashkan. A characterization of limiting functions arising in Mod-* convergence. Electron. Commun. Probab. 20 (2015), paper no. 79, 11 pp. doi:10.1214/ECP.v20-4381. https://projecteuclid.org/euclid.ecp/1465321006

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