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