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Diskr. Mat., 2017, Volume 29, Issue 1, Pages 136–155 (Mi dm1411)  

This article is cited in 1 scientific paper (total in 1 paper)

Limit theorems for the logarithm of the order of a random $A$-mapping

A. L. Yakymiv

Steklov Mathematical Institute of Russian Academy of Sciences, Moscow

Abstract: Let $\mathfrak S_n$ be a semigroup of mappings of a set $X$ with $n$ elements into itself, $A$ be some fixed subset of the set $N$ of natural numbers, and $V_n(A)$ be a set of mappings from $\mathfrak S_n$, with lengths of cycles belonging to $A$. The mappings from $V_n(A)$ are called $A$-mappings. We suppose that the set $A$ has an asymptotic density $\varrho>0$, and that $|k\colon k\leq n, k\in A, m-k\in A|/n\to\varrho^2$ as $n\to\infty$ uniformly over $m\in[n,Cn]$ for each constant $C>1$. A number $M(\alpha)$ of different elements in a set $\{\alpha, \alpha^2, \alpha^3,…\}$ is called an order of mapping $\alpha\in\mathfrak S_n$. Consider a random mapping $\sigma=\sigma_n(A)$ having uniform distribution on $V_n(A)$. In the present paper it is shown that random variable $\ln M(\sigma_n(A))$ is asymptotically normal with mean $l(n)=\sum_{k\in A(\sqrt{n})}\ln(k)/{k}$ and variance $\varrho\ln^3(n)/24$, where $A(t)=\{k\colon k\in A, k\leq t\}, t>0$. For the case $A=N$ this result was proved by B. Harris in 1973.

Keywords: random $A$-mappings, order of $A$-mapping, cyclic points, contours, trees, height of random\linebreak$A$-mapping, random $A$-permutations

Funding Agency Grant Number
Russian Foundation for Basic Research 14-01-00318_а
This study was supported by the Russian Foundation for Basic Research, grant 14-01-00318.


DOI: https://doi.org/10.4213/dm1411

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English version:
Discrete Mathematics and Applications, 2017, 27:5, 325–338

Bibliographic databases:

UDC: 519.212.2
Received: 28.07.2016
Revised: 21.11.2016

Citation: A. L. Yakymiv, “Limit theorems for the logarithm of the order of a random $A$-mapping”, Diskr. Mat., 29:1 (2017), 136–155; Discrete Math. Appl., 27:5 (2017), 325–338

Citation in format AMSBIB
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\by A.~L.~Yakymiv
\paper Limit theorems for the logarithm of the order of a random $A$-mapping
\jour Diskr. Mat.
\yr 2017
\vol 29
\issue 1
\pages 136--155
\mathnet{http://mi.mathnet.ru/dm1411}
\crossref{https://doi.org/10.4213/dm1411}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=3771057}
\elib{http://elibrary.ru/item.asp?id=28405141}
\transl
\jour Discrete Math. Appl.
\yr 2017
\vol 27
\issue 5
\pages 325--338
\crossref{https://doi.org/10.1515/dma-2017-0034}
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\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85031789597}


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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. A. L. Yakymiv, “On the order of random permutation with cycle weights”, Theory Probab. Appl., 63:2 (2018), 209–226  mathnet  crossref  crossref  isi  elib
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