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

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

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English version:
Theory of Probability and its Applications, 2010, 54:1, 114–128

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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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• http://mi.mathnet.ru/eng/tvp2499
• https://doi.org/10.4213/tvp2499
• http://mi.mathnet.ru/eng/tvp/v54/i1/p63

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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
2. A. L. Yakymiv, “Random $A$-permutations and Brownian motion”, Proc. Steklov Inst. Math., 282 (2013), 298–318
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
4. A. L. Yakymiv, “O raspredelenii tipa kratnogo stepennogo ryada, pravilno menyayuschegosya v granichnoi tochke”, Diskret. matem., 30:3 (2018), 141–158
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