A~problem in a~scheme of disposal of particles in cells, and cycles of random permutation

V. F. Kolčin

Teoriâ veroâtnostej i ee primeneniâ, Tome 16 (1971) no. 1, pp. 67-81

Résumé

We consider the uniform distribution on the set of all permutations of degree $n$. A random permutation from the set has a random number of cycles $\varkappa_n=\alpha_1+\dots+\alpha_n$ where $\alpha_r$ is the number of the cycles of length $r$. We arrange the cycles in accordance with their lengths and denote by $S_m$ the random variable equal to the length of the $m$-th cycle in the sequence. We prove that the distribution of $\alpha_1,\dots,\alpha_n$ coincides with the distribution of some random variables in a scheme of disposal of particles in cells. This permits us to reduce the investigation of $\alpha_1,\dots,\alpha_n$ and associated random variables $\varkappa_n$, $S_m$, $S_{\varkappa_n-m+1}$ to some problems on summation of independent identically distributed random variables. In this way we prove some limit theorems for random variables $\varkappa_n$, $S_m$ and $S_{\varkappa_n-m+1}$ analogous to those obtained in [3], [4].