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 Teor. Veroyatnost. i Primenen., 1971, Volume 16, Issue 1, Pages 148–156 (Mi tvp1982)

Short Communications

Random mappings with one attracting center

V. E. Stepanov

Moscow

Abstract: A random mapping $T$ of the set $\{a_0,a_1,…,a_n\}$ 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.

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English version:
Theory of Probability and its Applications, 1971, 16:1, 155–162

Bibliographic databases:

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

Citation in format AMSBIB
\Bibitem{Ste71} \by V.~E.~Stepanov \paper Random mappings with one attracting center \jour Teor. Veroyatnost. i Primenen. \yr 1971 \vol 16 \issue 1 \pages 148--156 \mathnet{http://mi.mathnet.ru/tvp1982} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=410842} \zmath{https://zbmath.org/?q=an:0239.60017} \transl \jour Theory Probab. Appl. \yr 1971 \vol 16 \issue 1 \pages 155--162 \crossref{https://doi.org/10.1137/1116013} 

• http://mi.mathnet.ru/eng/tvp1982
• http://mi.mathnet.ru/eng/tvp/v16/i1/p148

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. A. N. Timashev, “Random mappings of finite sets with a known number of components”, Theory Probab. Appl., 48:4 (2004), 741–751