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Sharp large deviations and concentration inequalities for the number of descents in a random permutation

Part of: Limit theorems Combinatorics

Published online by Cambridge University Press:  05 January 2024

Bernard Bercu ,
Michel Bonnefont  and
Adrien Richou Open the ORCID record for Adrien Richou [Opens in a new window]
Bernard Bercu*
Affiliation:
Université de Bordeaux, Institut de Mathématiques de Bordeaux
Michel Bonnefont*
Affiliation:
Université de Bordeaux, Institut de Mathématiques de Bordeaux
Adrien Richou*
Affiliation:
Université de Bordeaux, Institut de Mathématiques de Bordeaux
*
*Postal address: Université de Bordeaux, Institut de Mathématiques de Bordeaux, UMR CNRS 5251, 351 Cours de la Libération, 33405 Talence cedex, France.
*Postal address: Université de Bordeaux, Institut de Mathématiques de Bordeaux, UMR CNRS 5251, 351 Cours de la Libération, 33405 Talence cedex, France.
*Postal address: Université de Bordeaux, Institut de Mathématiques de Bordeaux, UMR CNRS 5251, 351 Cours de la Libération, 33405 Talence cedex, France.
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Abstract

The goal of this paper is to go further in the analysis of the behavior of the number of descents in a random permutation. Via two different approaches relying on a suitable martingale decomposition or on the Irwin–Hall distribution, we prove that the number of descents satisfies a sharp large-deviation principle. A very precise concentration inequality involving the rate function in the large-deviation principle is also provided.

Keywords

Large deviationsconcentration inequalitiesrandom permutations

MSC classification

Primary: 60F10: Large deviations
Secondary: 05A05: Permutations, words, matrices
Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 24
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

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