Limit distributions of certain characteristics of random mappings

V. E. Stepanov

Teoriâ veroâtnostej i ee primeneniâ, Tome 14 (1969) no. 4, pp. 639-653

Résumé

A random mapping of a set $X_n$ of $n$ elements into $X_n$ being under consideration, the distributions of its various characteristics such as the number of components, the number of points of different orders, the number of trees of one or another size etc. are studied. Here is a typical example of the results obtained: let $\zeta^{(n)}(s)$ be the number of points, the order of which is greater than $s$ and $n^{-1/2}s\to\alpha$, $0<\alpha<\infty$, as $n\to\infty$; then the random variable $n^{-1/2}\zeta^{(n)}(s)$ has the limit distribution with the Laplace transform $\Psi_\alpha(t)$ defined by $$ \Psi_\alpha(t)=\sqrt{2\pi}\frac1{2\pi i}\int_{1-i\infty}^{1+i\infty}\exp\{E(\alpha\sqrt{p^2+2t})-E(\alpha p)\}\cdot e^{p^2/2}\,dp $$ where $E(p)=\int_p^\infty x^{-1}e^{-x}\,dx$.