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Total variation distance and the Erdős-Turán law for random permutations with polynomially growing cycle weights

Research output: Contribution to Journal/Magazine › Journal article › peer-review

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Total variation distance and the Erdős-Turán law for random permutations with polynomially growing cycle weights. / Storm, Julia; Zeindler, Dirk.
In: Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, Vol. 52, No. 4, 17.11.2016, p. 1614-1640.

Research output: Contribution to Journal/Magazine › Journal article › peer-review

Harvard

Storm, J & Zeindler, D 2016, 'Total variation distance and the Erdős-Turán law for random permutations with polynomially growing cycle weights', Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, vol. 52, no. 4, pp. 1614-1640. https://doi.org/10.1214/15-AIHP692

APA

Storm, J., & Zeindler, D. (2016). Total variation distance and the Erdős-Turán law for random permutations with polynomially growing cycle weights. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 52(4), 1614-1640. https://doi.org/10.1214/15-AIHP692

Vancouver

Storm J, Zeindler D. Total variation distance and the Erdős-Turán law for random permutations with polynomially growing cycle weights. Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques. 2016 Nov 17;52(4):1614-1640. doi: 10.1214/15-AIHP692

Author

Storm, Julia ; Zeindler, Dirk. / Total variation distance and the Erdős-Turán law for random permutations with polynomially growing cycle weights. In: Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques. 2016 ; Vol. 52, No. 4. pp. 1614-1640.

Bibtex

@article{546522756eba480dad4aa8e8177e8e94,
title = "Total variation distance and the Erd{\H o}s-Tur{\'a}n law for random permutations with polynomially growing cycle weights",
abstract = "We study the model of random permutations of $n$ objects with polynomially growing cycle weights, which was recently considered by Ercolani and Ueltschi, among others. Using saddle-point analysis, we prove that the total variation distance between the process which counts the cycles of size $1, 2, ..., b$ and a process $(Z_1, Z_2, ..., Z_b)$ of independent Poisson random variables converges to $0$ if and only if $b=o(\ell)$ where $\ell$ denotes the length of a typical cycle in this model. By means of this result, we prove a central limit theorem for the order of a permutation and thus extend the Erd\H{o}s-Tur\'an Law to this measure. Furthermore, we prove a Brownian motion limit theorem for the small cycles.",
keywords = "math.PR, 60C05, 60B15, 60F17",
author = "Julia Storm and Dirk Zeindler",
year = "2016",
month = nov,
day = "17",
doi = "10.1214/15-AIHP692",
language = "English",
volume = "52",
pages = "1614--1640",
journal = "Annales de l'Institut Henri Poincar{\'e} (B) Probabilit{\'e}s et Statistiques",
issn = "0246-0203",
publisher = "Institute of Mathematical Statistics",
number = "4",

}

RIS

TY - JOUR

T1 - Total variation distance and the Erdős-Turán law for random permutations with polynomially growing cycle weights

AU - Storm, Julia

AU - Zeindler, Dirk

PY - 2016/11/17

Y1 - 2016/11/17

N2 - We study the model of random permutations of $n$ objects with polynomially growing cycle weights, which was recently considered by Ercolani and Ueltschi, among others. Using saddle-point analysis, we prove that the total variation distance between the process which counts the cycles of size $1, 2, ..., b$ and a process $(Z_1, Z_2, ..., Z_b)$ of independent Poisson random variables converges to $0$ if and only if $b=o(\ell)$ where $\ell$ denotes the length of a typical cycle in this model. By means of this result, we prove a central limit theorem for the order of a permutation and thus extend the Erd\H{o}s-Tur\'an Law to this measure. Furthermore, we prove a Brownian motion limit theorem for the small cycles.

AB - We study the model of random permutations of $n$ objects with polynomially growing cycle weights, which was recently considered by Ercolani and Ueltschi, among others. Using saddle-point analysis, we prove that the total variation distance between the process which counts the cycles of size $1, 2, ..., b$ and a process $(Z_1, Z_2, ..., Z_b)$ of independent Poisson random variables converges to $0$ if and only if $b=o(\ell)$ where $\ell$ denotes the length of a typical cycle in this model. By means of this result, we prove a central limit theorem for the order of a permutation and thus extend the Erd\H{o}s-Tur\'an Law to this measure. Furthermore, we prove a Brownian motion limit theorem for the small cycles.

KW - math.PR

KW - 60C05, 60B15, 60F17

U2 - 10.1214/15-AIHP692

DO - 10.1214/15-AIHP692

M3 - Journal article

VL - 52

SP - 1614

EP - 1640

JO - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques

JF - Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques

SN - 0246-0203

IS - 4

ER -