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Information-theoretic convergence of extreme values to the Gumbel distribution

Part of: Communication, information Limit theorems Sufficiency and information

Published online by Cambridge University Press:  21 June 2023

Oliver Johnson Open the ORCID record for Oliver Johnson [Opens in a new window]
Oliver Johnson*
Affiliation:
University of Bristol
*
*Postal address: School of Mathematics, University of Bristol, Fry Building, Woodland Road, Bristol, BS8 1UG, UK. Email: O.Johnson@bristol.ac.uk
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Abstract

We show how convergence to the Gumbel distribution in an extreme value setting can be understood in an information-theoretic sense. We introduce a new type of score function which behaves well under the maximum operation, and which implies simple expressions for entropy and relative entropy. We show that, assuming certain properties of the von Mises representation, convergence to the Gumbel distribution can be proved in the strong sense of relative entropy.

Keywords

entropyextreme value theoryinformation theoryvon Mises representation

MSC classification

Primary: 60F99: None of the above, but in this section
Secondary: 62B10: Information-theoretic topics 94A15: Information theory, general
Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 1 , March 2024 , pp. 244 - 254
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of Applied Probability Trust

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