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 Teor. Veroyatnost. i Primenen.: Year: Volume: Issue: Page: Find

 Teor. Veroyatnost. i Primenen., 1969, Volume 14, Issue 1, Pages 64–77 (Mi tvp1117)

On the distribution of the number of vertices in strata of a random tree

V. E. Stepanov

Moscow

Abstract: The set of trees with $m+1$ distinguishable vertices one of which is taken as a root is considered. In every tree all the vertices are distributed in strata with respect to the root according to the lengths of paths which connect them to the root.
Let $\zeta_{m,j}$ be the number of vertices in the $j$-th stratum of a tree chosen at random.
We prove that if $m$ and $j\to\infty$ so that $j/\sqrt m\to\alpha$, $0<\alpha_1\le\alpha<\alpha_2<\infty$, then the distributions of random variables $\zeta_{m,j}/\sqrt m$ converge to a limit distribution. Explicit expressions for moments and the density of the limit distribution are found.

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English version:
Theory of Probability and its Applications, 1969, 14:1, 65–78

Bibliographic databases:

Citation: V. E. Stepanov, “On the distribution of the number of vertices in strata of a random tree”, Teor. Veroyatnost. i Primenen., 14:1 (1969), 64–77; Theory Probab. Appl., 14:1 (1969), 65–78

Citation in format AMSBIB
\Bibitem{Ste69} \by V.~E.~Stepanov \paper On the distribution of the number of vertices in strata of a~random tree \jour Teor. Veroyatnost. i Primenen. \yr 1969 \vol 14 \issue 1 \pages 64--77 \mathnet{http://mi.mathnet.ru/tvp1117} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=246344} \zmath{https://zbmath.org/?q=an:0196.21001|0172.21902} \transl \jour Theory Probab. Appl. \yr 1969 \vol 14 \issue 1 \pages 65--78 \crossref{https://doi.org/10.1137/1114007} 

• http://mi.mathnet.ru/eng/tvp1117
• http://mi.mathnet.ru/eng/tvp/v14/i1/p64

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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. Yu. L. Pavlov, “Limit theorems for the number of trees of a given size in a random forest”, Math. USSR-Sb., 32:3 (1977), 335–345
2. A. M. Zubkov, A. A. Serov, “Images of subset of finite set under iterations of random mappings”, Discrete Math. Appl., 25:3 (2015), 179–185
3. A. A. Serov, “Images of a finite set under iterations of two random dependent mappings”, Discrete Math. Appl., 26:3 (2016), 175–181