The cyclic structure of random permutations

V. F. Kolchin ; V. P. Chistyakov

Matematičeskie zametki, Tome 18 (1975) no. 6, pp. 929-938.

Résumé

Let $\alpha_r$ denote the number of cycles of length $r$ in a random permutation, taking its values with equal probability from among the set $S_n$ of all permutations of length $n$. In this paper we study the limiting behavior of linear combinations of random permutations $\alpha_1,\dots,\alpha_r$ having the form $$ \zeta_{n,r}=C_{r1}\alpha_1+\dots+C_{rr}\alpha_r $$ in the case when $n,r\to\infty$. We shall show that the class of limit distributions for $\zeta_{n,r}$ as $n,r\to\infty$ and $r\ln r/n\to0$ coincides with the class of unbounded divisible distributions. For the random variables $\eta_{n,r}=\alpha_1+2\alpha_2+\dots+r\alpha_r$, equal to the number of elements in the permutation contained in cycles of length not exceeding $r$, we find limit distributions of the form $r\ln r/n\to0$ и $r=\gamma n$, $0\gamma1$.