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 Teor. Veroyatnost. i Primenen., 2018, Volume 63, Issue 2, Pages 260–283 (Mi tvp5147)

On the order of random permutation with cycle weights

A. L. Yakymiv

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Let $\operatorname{Ord}(\tau)$ be the order of an element $\tau$ in the group $S_n$ of permutations of an $n$-element set $X$. The present paper is concerned with the so-called general parametric model of a random permutation; according to this model an arbitrary fixed permutation $\tau$ from $S_n$ is observed with the probability $\theta_1^{u_1}\dotsb\theta_n^{u_n}/H(n)$, where $u_i$ is the number of cycles of length $i$ of the permutation $\tau$, $\{\theta_i, i\in \mathbf{N}\}$ are some nonnegative parameters (the weights of cycles of length $i$ of the permutation $\tau$), and $H(n)$ is the corresponding normalizing factor. We assume that an arbitrary permutation $\tau_n$ has such a distribution. The function $p(n)=H(n)/n!$ is assumed to be $\mathrm{RO}$-varying at infinity with the lower index exceeding $-1$ (in particular, it can vary regularly), and the sequence $\{\theta_i, i\in \mathbf N\}$ is bounded. Under these assumptions it is shown that the random variable $\ln\operatorname{Ord}(\tau_n)$ is asymptotically normal with mean $\sum_{k=1}^n\theta_k\ln (k)/k$ and variance $\sum_{k=1}^n\theta_k\ln^2(k)/k$. In particular, this scheme subsumes the class of random $A$-permutations (i.e., when $\theta_i=\chi\{i\in A\}$), where $A$ is an arbitrary fixed subset of the positive integers. This scheme also includes the Ewens model of random permutation, where $\theta_i\equiv\theta>0$ for any $i\in\mathbf N$. The limit theorem we prove here extends some previous results for these schemes. In particular, with $\theta_i\equiv1$ for any $i\in\mathbf N$, the result just mentioned implies the well-known Erdős–Turán limit theorem.

Keywords: random permutation with cycle weights, random $A$-permutation, random permutation in the Ewens mode, order of random permutation, regularly varying function, $\mathrm{RO}$-varying function.

 Funding Agency Grant Number Russian Academy of Sciences - Federal Agency for Scientific Organizations PRAS-18-01 This work was supported by Program of the Presidium of the Russian Academy of Sciences no. 01 “Fundamental Mathematics and Its Applications” under grant PRAS-18-0.

DOI: https://doi.org/10.4213/tvp5147

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English version:
Theory of Probability and its Applications, 2018, 63:2, 209–226

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Accepted:22.11.2017

Citation: A. L. Yakymiv, “On the order of random permutation with cycle weights”, Teor. Veroyatnost. i Primenen., 63:2 (2018), 260–283; Theory Probab. Appl., 63:2 (2018), 209–226

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/tvp5147
• https://doi.org/10.4213/tvp5147
• http://mi.mathnet.ru/eng/tvp/v63/i2/p260

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This publication is cited in the following articles:
1. A. L. Yakymiv, “Asymptotics with remainder term for moments of the total cycle number of random $A$-permutation”, Discrete Math. Appl., 31:1 (2021), 51–60
2. A. L. Yakymiv, “Size distribution of the largest component of a random $A$-mapping”, Discrete Math. Appl., 31:2 (2021), 145–153
3. A. L. Yakymiv, “Abelian theorem for the regularly varying measure and its density in orthant”, Theory Probab. Appl., 64:3 (2019), 385–400
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