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 Diskr. Mat., 2011, Volume 23, Issue 1, Pages 21–27 (Mi dm1127)

Limit theorems for the joint distribution of component sizes of a random mapping with a known number of components

A. N. Timashov

Abstract: We consider the mapping $C_{N,n}$ of a set with $n$ numbered elements into itself, which has $N\le n$ connected components and is uniformly distributed on the set of all such mappings. We denote the number of such mappings by $a(n, N)$. In addition to the known estimates we derive some new estimates of the number $a(n, N)$ under the condition that $n\to\infty$ and $N=N(n)$.
Let $\eta_1,…,\eta_N$ be the sizes of connected components of the random mapping $C_{N,n}$, numbered in one of the $N!$ possible ways. We obtain limit theorems estimating the distribution of the random vector $(\eta_1,…,\eta_N)$ as $n,N\to\infty$ including the domain of large deviations. A new asymptotic estimate of the local probabilities for a sum of independent identically distributed random variables which determine the corresponding generalised allocation scheme is obtained.

DOI: https://doi.org/10.4213/dm1127

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English version:
Discrete Mathematics and Applications, 2011, 21:1, 39–46

Bibliographic databases:

UDC: 519.24

Citation: A. N. Timashov, “Limit theorems for the joint distribution of component sizes of a random mapping with a known number of components”, Diskr. Mat., 23:1 (2011), 21–27; Discrete Math. Appl., 21:1 (2011), 39–46

Citation in format AMSBIB
\Bibitem{Tim11} \by A.~N.~Timashov \paper Limit theorems for the joint distribution of component sizes of a~random mapping with a~known number of components \jour Diskr. Mat. \yr 2011 \vol 23 \issue 1 \pages 21--27 \mathnet{http://mi.mathnet.ru/dm1127} \crossref{https://doi.org/10.4213/dm1127} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2830694} \elib{http://elibrary.ru/item.asp?id=20730369} \transl \jour Discrete Math. Appl. \yr 2011 \vol 21 \issue 1 \pages 39--46 \crossref{https://doi.org/10.1515/DMA.2011.003} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79953302017} 

• http://mi.mathnet.ru/eng/dm1127
• https://doi.org/10.4213/dm1127
• http://mi.mathnet.ru/eng/dm/v23/i1/p21

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