|
This article is cited in 5 scientific papers (total in 5 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.
Received: 13.06.2017 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
Linking options:
https://www.mathnet.ru/eng/tvp5147https://doi.org/10.4213/tvp5147 https://www.mathnet.ru/eng/tvp/v63/i2/p260
|
Statistics & downloads: |
Abstract page: | 548 | Full-text PDF : | 132 | References: | 57 | First page: | 9 |
|