On the Order of Random Permutation with Cycle Weights
Arsen Lubomirovich Yakymiv · Theory of Probability and Its Applications · 2018
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 Erdös--Turán limit theorem.