The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles

Ashkan Nikeghbali; Dirk Zeindler

Annales de l'I.H.P. Probabilités et statistiques (2013)

  • Volume: 49, Issue: 4, page 961-981
  • ISSN: 0246-0203

Abstract

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The goal of this paper is to analyse the asymptotic behaviour of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables and to establish a central limit theorem for the total number of cycles. Our methods allow us to obtain an asymptotic estimate of the characteristic functions of the different random vectors of interest together with an error estimate, thus having a control on the speed of convergence. In fact we are able to prove a finer convergence for the total number of cycles, namely mod-Poisson convergence. From there we apply previous results on mod-Poisson convergence to obtain Poisson approximation for the total number of cycles as well as large deviations estimates.

How to cite

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Nikeghbali, Ashkan, and Zeindler, Dirk. "The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles." Annales de l'I.H.P. Probabilités et statistiques 49.4 (2013): 961-981. <http://eudml.org/doc/271946>.

@article{Nikeghbali2013,
abstract = {The goal of this paper is to analyse the asymptotic behaviour of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables and to establish a central limit theorem for the total number of cycles. Our methods allow us to obtain an asymptotic estimate of the characteristic functions of the different random vectors of interest together with an error estimate, thus having a control on the speed of convergence. In fact we are able to prove a finer convergence for the total number of cycles, namely mod-Poisson convergence. From there we apply previous results on mod-Poisson convergence to obtain Poisson approximation for the total number of cycles as well as large deviations estimates.},
author = {Nikeghbali, Ashkan, Zeindler, Dirk},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {symmetric group; weighted probability measure; cycle counts; total number cycles; mod-Poisson convergence; Poisson approximation},
language = {eng},
number = {4},
pages = {961-981},
publisher = {Gauthier-Villars},
title = {The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles},
url = {http://eudml.org/doc/271946},
volume = {49},
year = {2013},
}

TY - JOUR
AU - Nikeghbali, Ashkan
AU - Zeindler, Dirk
TI - The generalized weighted probability measure on the symmetric group and the asymptotic behavior of the cycles
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2013
PB - Gauthier-Villars
VL - 49
IS - 4
SP - 961
EP - 981
AB - The goal of this paper is to analyse the asymptotic behaviour of the cycle process and the total number of cycles of weighted and generalized weighted random permutations which are relevant models in physics and which extend the Ewens measure. We combine tools from combinatorics and complex analysis (e.g. singularity analysis of generating functions) to prove that under some analytic conditions (on relevant generating functions) the cycle process converges to a vector of independent Poisson variables and to establish a central limit theorem for the total number of cycles. Our methods allow us to obtain an asymptotic estimate of the characteristic functions of the different random vectors of interest together with an error estimate, thus having a control on the speed of convergence. In fact we are able to prove a finer convergence for the total number of cycles, namely mod-Poisson convergence. From there we apply previous results on mod-Poisson convergence to obtain Poisson approximation for the total number of cycles as well as large deviations estimates.
LA - eng
KW - symmetric group; weighted probability measure; cycle counts; total number cycles; mod-Poisson convergence; Poisson approximation
UR - http://eudml.org/doc/271946
ER -

References

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