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

This article is cited in 3 scientific papers (total in 3 papers)

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.


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

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Received: 13.06.2017

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
\by A.~L.~Yakymiv
\paper On the order of random permutation with cycle weights
\jour Teor. Veroyatnost. i Primenen.
\yr 2018
\vol 63
\issue 2
\pages 260--283
\jour Theory Probab. Appl.
\yr 2018
\vol 63
\issue 2
\pages 209--226

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    This publication is cited in the following articles:
    1. A. L. Yakymiv, “Asimptotika momentov chisla tsiklov sluchainoi $A$-podstanovki s ostatochnym chlenom”, Diskret. matem., 31:3 (2019), 114–127  mathnet  crossref  mathscinet
    2. A. L. Yakymiv, “Raspredelenie ob'ema naibolshei komponenty sluchainogo $A$-otobrazheniya”, Diskret. matem., 31:4 (2019), 116–127  mathnet  crossref
    3. A. L. Yakymiv, “Abelian theorem for the regularly varying measure and its density in orthant”, Theory Probab. Appl., 64:3 (2019), 385–400  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
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