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# Discrete Mathematics and Applications

Editor-in-Chief: Zubkov, Andrei

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CiteScore 2016: 0.16

SCImago Journal Rank (SJR) 2016: 0.231
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# Images of a finite set under iterations of two random dependent mappings

Aleksandr A. Serov
Published Online: 2016-07-15 | DOI: https://doi.org/10.1515/dma-2016-0015

## Abstract

Let $\mathcal{N}$ be a set of N elements and $\left({F}_{1},{G}_{1}\right),\left({F}_{2},{G}_{2}\right),\dots$ be a sequence of independent pairs of random dependent mappings $\mathcal{N}\to \mathcal{N}$ such that Fk and Gk are random equiprobable mappings and $\mathbf{P}\left\{{F}_{k}\left(x\right)={G}_{k}\left(x\right)\right\}=\alpha$ for all $x\in \mathcal{N}$ and k = 1, 2, … For a subset ${S}_{0}\subset \mathcal{N},\phantom{\rule{0.056em}{0ex}}|{S}_{0}|=n$, we consider a sequences of its images ${S}_{k}={F}_{k}\left(\dots {F}_{2}\left({F}_{1}\left({S}_{0}\right)\right)\dots \right)$, ${T}_{k}={G}_{k}\left(\dots {G}_{2}\left({G}_{1}\left({S}_{0}\right)\right)\dots \right)$, k = 1, 2 …, and a sequences of their unions ${S}_{k}\cup {T}_{k}$ and intersections ${S}_{k}\cap {T}_{k}$, k = 1, 2 … We obtain two-sided inequalities for $\mathbf{M}|{S}_{k}\cup {T}_{k}|$ and $\mathbf{M}|{S}_{k}\cap {T}_{k}|$ such that upper and lower bounds are asymptotically equivalent if $N,n,k\to \mathrm{\infty }$, $nk=o\left(N\right)$ and $\alpha =O\left(\frac{1}{N}\right)$.

Note: Originally published in Diskretnaya Matematika (2015) 27, ${\mathrm{N}}^{\underset{_}{\mathrm{o}}}$4, 133–140 (in Russian).

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Published Online: 2016-07-15

Published in Print: 2016-07-01

Funding Source: Russian Science Foundation

Award identifier / Grant number: 14-50-00005

This work was supported by the Russian Science Foundation under grant no. 14-50-00005

Citation Information: Discrete Mathematics and Applications, Volume 26, Issue 3, Pages 175–181, ISSN (Online) 1569-3929, ISSN (Print) 0924-9265,

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