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Mat. Zametki, 2015, Volume 97, Issue 3, Pages 462–470 (Mi mz9094)  

On the Number of Components of Fixed Size in a Random $A$-Mapping

A. L. Yakymiv

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Let $\mathfrak S_n$ be the semigroup of mappings of a set of $n$ elements into itself, let $A$ be a fixed subset of the set of natural numbers $\mathbb N$, and let $V_n(A)$ be the set of mappings from $\mathfrak S_n$ for which the sizes of the contours belong to the set $A$. Mappings from $V_n(A)$ are usually called $A$-mappings. Consider a random mapping $\sigma_n$ uniformly distributed on $V_n(A)$. It is assumed that the set $A$ possesses asymptotic density $\varrho$, including the case $\varrho=0$. Let $\xi_{in}$ be the number of connected components of a random mapping $\sigma_n$ of size $i\in\mathbb N$. For a fixed integer $b\in\mathbb N$, as $n\to\infty$, the asymptotic behavior of the joint distribution of random variables $\xi_{1n},\xi_{2n},…,\xi_{bn}$ is studied. It is shown that this distribution weakly converges to the joint distribution of independent Poisson random variables $\eta_1,\eta_2,…,\eta_b$ with some parameters $\lambda_i=\mathsf E\eta_{i}$, $i\in\mathbb N$.

Keywords: random $A$-mapping, Poisson random variable, asymptotic behavior of the joint distribution of random variables, regularly/slowly varying function in the sense of Karamata, Stirling's formula.

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00318-a
This work was supported by the Russian Foundation for Basic Research (grant no. 14-01-00318-a).


DOI: https://doi.org/10.4213/mzm9094

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English version:
Mathematical Notes, 2015, 97:3, 468–475

Bibliographic databases:

UDC: 519.174
Received: 11.06.2013
Revised: 27.10.2014

Citation: A. L. Yakymiv, “On the Number of Components of Fixed Size in a Random $A$-Mapping”, Mat. Zametki, 97:3 (2015), 462–470; Math. Notes, 97:3 (2015), 468–475

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