Random partitions of sets

V. N. Sachkov

Teoriâ veroâtnostej i ee primeneniâ, Tome 19 (1974) no. 1, pp. 187-194

Résumé

Let $\mathfrak A_m$ be a set of $m$ elements and each its partition into subsets be equiprobable. Let $\xi_l$ be the number of subsets of power $l$ in the random partition. Then the vector $$ ((\xi_{i_1}-\lambda_{i_1})/\sqrt{\lambda_{i_1}},\dots,(\xi_{i_k}-\lambda_{i_k})/\sqrt{\lambda_{i_k}}), $$ where $\lambda_l=r^l/l!$, $r$ being the unique real root of the equation $re^r=m$, is shown to be asymptotically normal as $m\to\infty$ with unit variances and independent components. The limit distributions of $\mu_m$ and $\nu_m$ are studied, $\mu_m$ $(\nu_m)$ being the greatest (least) power in the random partition of $\mathfrak A_m$. The first is shown to be close to a double exponential distribution in a neighbourhood of point $er$, the second to be the degenerate distribution with the unit mass at point 1.