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

This article is cited in 1 scientific paper (total in 1 paper)

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:

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

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}


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    Citing articles on Google Scholar: Russian citations, English citations
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    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  mathnet  crossref  crossref  mathscinet  zmath  isi
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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