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Diskr. Mat., 2015, Volume 27, Issue 4, Pages 133–140 (Mi dm1352)  

This article is cited in 3 scientific papers (total in 3 papers)

Images of a finite set under iterations of two random dependent mappings

A. A. Serov

Steklov Mathematical Institute of Russian Academy of Sciences

Abstract: Let $\mathcal{N}$ be a set of $N$ elements and $(F_1,G_1),(F_2,G_2),\ldots$ be a sequence of independent pairs of random dependent mappings $\mathcal{N}\to\mathcal{N}$ such that $F_k$ and $G_k$ are random equiprobable mappings and $\mathbf{P}\{F_k(x)=G_k(x)\}=\alpha$ for all $x\in \mathcal{N}$ and $k=1,2,\ldots$ For a subset $S_0\subset \mathcal{N}, |S_0|=n$, we consider a sequences of its images $S_k=F_k(\ldots F_2(F_1(S_0))\ldots)$, $T_k=G_k(\ldots G_2(G_1(S_0))\ldots)$, $k=1,2\ldots$, and a sequences of their unions $S_k\cup T_k$ and intersections $S_k\cap T_k$, $k=1,2\ldots$ 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 $N,n,k\to\infty$, $nk=o(N)$ and $\alpha=O(\tfrac1N)$.

Keywords: random mappings of finite sets, joint distributions, iterations of random mappings, Markov chain

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


DOI: https://doi.org/10.4213/dm1352

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English version:
Discrete Mathematics and Applications, 2016, 26:3, 175–181

Bibliographic databases:

Document Type: Article
UDC: 519.212.2+519.213.21
Received: 30.10.2015

Citation: A. A. Serov, “Images of a finite set under iterations of two random dependent mappings”, Diskr. Mat., 27:4 (2015), 133–140; Discrete Math. Appl., 26:3 (2016), 175–181

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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. M. Zubkov, A. A. Serov, “Limit theorem for the size of an image of subset under compositions of random mappings”, Discrete Math. Appl., 28:2 (2018), 131–138  mathnet  crossref  crossref  isi  elib
    2. A. M. Zubkov, V. O. Mironkin, “Raspredelenie dliny otrezka aperiodichnosti v grafe $k$-kratnoi iteratsii sluchainogo ravnoveroyatnogo otobrazheniya”, Matem. vopr. kriptogr., 8:4 (2017), 63–74  mathnet  crossref  mathscinet  elib
    3. A. M. Zubkov, A. A. Serov, “Estimates of the mean size of the subset image under composition of random mappings”, Discrete Math. Appl., 28:5 (2018), 331–338  mathnet  crossref  crossref  isi  elib
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