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Teoriya Veroyatnostei i ee Primeneniya, 1969, Volume 14, Issue 4, Pages 639–653
(Mi tvp1446)
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This article is cited in 39 scientific papers (total in 39 papers)
Limit distributions of certain characteristics of random mappings
V. E. Stepanov Moscow
Abstract:
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 $E(p)=\int_p^\infty x^{-1}e^{-x}\,dx$.
Received: 28.03.1968
Citation:
V. E. Stepanov, “Limit distributions of certain characteristics of random mappings”, Teor. Veroyatnost. i Primenen., 14:4 (1969), 639–653; Theory Probab. Appl., 14:4 (1969), 616–626
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