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Teor. Veroyatnost. i Primenen., 2009, Volume 54, Issue 1, Pages 63–79 (Mi tvp2499)  

This article is cited in 4 scientific papers (total in 4 papers)

Limit Theorem for the Middle Members of Ordered Cycle Lengths in Random $A$-Permutations

A. L. Yakymiv

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: In this article, random permutation $\tau_n$ is considered uniformly distributed on the set of all permutations with degree $n$ and with cycle lengths from fixed set $A$ (so-called $A$-permutations). Let $\zeta_n$ be the general number of cycles and $\eta_n(1)\leq\eta_n(2)\leq\cdots\leq\eta_n(\zeta_n)$ be the ordered cycle lengths in a random permutation $\tau_n$. The central limit theorem is obtained here for the middle members of this sequence, i.e., for random variables $\eta_n(m)$ with numbers $m=\alpha\log n+o(\sqrt{\log n})$ as $n\to\infty$ for fixed $\alpha\in(0,\sigma)$ and for some class of the sets $A$ with positive asymptotic density $\sigma$. The basic approach to the proof is the new three-dimensional Tauberian theorem. Asymptotic behavior of extreme left and extreme right members of this sequence was investigated earlier by the author.

Keywords: random $A$-permutation, ordered cycle lengths of permutation, Tauberian theorem

DOI: https://doi.org/10.4213/tvp2499

Full text: PDF file (193 kB)
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English version:
Theory of Probability and its Applications, 2010, 54:1, 114–128

Bibliographic databases:

Received: 01.12.2006
Revised: 31.10.2007

Citation: A. L. Yakymiv, “Limit Theorem for the Middle Members of Ordered Cycle Lengths in Random $A$-Permutations”, Teor. Veroyatnost. i Primenen., 54:1 (2009), 63–79; Theory Probab. Appl., 54:1 (2010), 114–128

Citation in format AMSBIB
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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, “A limit theorem for the logarithm of the order of a random $A$-permutation”, Discrete Math. Appl., 20:3 (2010), 247–275  mathnet  crossref  crossref  mathscinet  zmath  elib  elib
    2. A. L. Yakymiv, “Random $A$-permutations and Brownian motion”, Proc. Steklov Inst. Math., 282 (2013), 298–318  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    3. A. M. Shaiduk, S. A. Ostanin, G. A. Semenov, “Reiting kak sledstvie printsipa maksimuma entropii”, Mezhdunar. nauch.-issled. zhurn., 2016, no. 8-3(50), 158–164  mathnet  crossref
    4. A. L. Yakymiv, “O raspredelenii tipa kratnogo stepennogo ryada, pravilno menyayuschegosya v granichnoi tochke”, Diskret. matem., 30:3 (2018), 141–158  mathnet  crossref  elib
  • Теория вероятностей и ее применения Theory of Probability and its Applications
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