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Teoriya Veroyatnostei i ee Primeneniya, 1971, Volume 16, Issue 1, Pages 148–156
(Mi tvp1982)
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This article is cited in 31 scientific papers (total in 31 papers)
Short Communications
Random mappings with one attracting center
V. E. Stepanov Moscow
Abstract:
A random mapping $T$ of the set $\{a_0,a_1,\dots,a_n\}$ into itself is determined by the following requirements: 1) images of the points $a_i$, $0\le i\le n$, are chosen at random and independently; 2) for any $i$
$$
\mathbf P(Ta_i=a_0)=\lambda/(n+\lambda),\quad\lambda\ge1;\quad\mathbf P(Ta_i=a_j)=1/(n+\lambda),\quad1\le j\le n.
$$
Vertex $a_0$ is called an attracting center of weight $\lambda$. The graph component of mapping $T$ containing the center, the cycle belonging to it and all its vertices are called principal, and all the rest components, cycles and vertices are called free.
Limit distributions of various characteristics of random mappings with one attracting center of weight $\lambda$ are studied in this paper. For example, it is shown that if $\lambda$ varies an $n\to\infty$ so that $\lambda/\sqrt n\to\infty$ but $\lambda/n\to0$ the distribution of the random variable $\lambda^2\xi_n(\lambda)/n^2$ where $\xi_n(\lambda)$ is the number of free vertices converges to the $\chi^2$-distribution with one degree of freedom.
Received: 22.12.1969
Citation:
V. E. Stepanov, “Random mappings with one attracting center”, Teor. Veroyatnost. i Primenen., 16:1 (1971), 148–156; Theory Probab. Appl., 16:1 (1971), 155–162
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