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On the rate of growth of random analytic functions, with an application to linear dynamics

Part of: Series expansions General theory of linear operators Entire and meromorphic functions, and related topics

Published online by Cambridge University Press:  28 August 2025

Kevin Agneessens Open the ORCID record for Kevin Agneessens [Opens in a new window]  and
Karl Grosse-Erdmann Open the ORCID record for Karl Grosse-Erdmann [Opens in a new window]
Kevin Agneessens
Affiliation:
Université de Mons , Département de Mathématique, 20 Place du Parc, 7000 Mons, Belgium e-mail: agneessens.kevin@gmail.com
Karl Grosse-Erdmann*
Affiliation:
Université de Mons , Département de Mathématique, 20 Place du Parc, 7000 Mons, Belgium e-mail: agneessens.kevin@gmail.com
*
e-mail: kg.grosse-erdmann@umons.ac.be
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Abstract

We obtain Wiman–Valiron type inequalities for random entire functions and for random analytic functions on the unit disk that improve a classical result of Erdős and Rényi and recent results of Kuryliak and Skaskiv. Our results are then applied to linear dynamics: we obtain rates of growth, outside some exceptional set, for analytic functions that are frequently hypercyclic for an arbitrary chaotic weighted backward shift.

Keywords

Wiman–Valiron inequalityLévy’s phenomenonsubgaussian random variablefrequently hypercyclic entire functionrate of growth

MSC classification

Primary: 30B20: Random power series 30D20: Entire functions, general theory 47A16: Cyclic vectors, hypercyclic and chaotic operators

Information

Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 21
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The first author was a Research Fellow of the Fonds de la Recherche Scientifique – FNRS. The second author was supported by the Fonds de la Recherche Scientifique – FNRS under Grant n° CDR J.0078.21.

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