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 Teor. Veroyatnost. i Primenen., 2007, Volume 52, Issue 1, Pages 69–83 (Mi tvp5)

Limit theorem for the general number of cycles in a random $A$-permutation

A. L. Yakymiv

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Let $S_n$ be the symmetric group of all permutations of degree $n, A$ be some nonempty subset of the set of natural numbers $N$, and let $T_n=T_n(A)$ be the set of all permutations from $S_n$ with cycle lengths from $A$. The permutations from $T_n$ are called $A$-permutations. Let $\zeta_n$ be the general number of cycles in a random permutation uniformly distributed on $T_n$. In this paper, we find the way to prove the limit theorem for $\zeta_n$ starting with the asymptotics of $|T_n|$. The limit theorem obtained here is new in a number of cases when the asymptotics of $|T_n|$ is known but the limit theorem for $\zeta_n$ has not yet been proven by other methods. As has been noted by the author, $|T_n|/n!$ is the Karamata regularly varying function with index $\sigma-1$, where $\sigma>0$ is the density of the set $A$, in a number of papers of different authors. Proof of the limit theorem for $\zeta_n$ is the main goal of this paper, assuming none of the additional restrictions typical of previous investigations.

Keywords: asymptotic density of the set $A$, logarithmic density of the set $A$, random $A$-permutations, general number of cycles in random $A$-permutation, regularly varying functions, slowly varying functions, Tauberian theorem.

DOI: https://doi.org/10.4213/tvp5

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English version:
Theory of Probability and its Applications, 2008, 52:1, 133–146

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Revised: 06.09.2006

Citation: A. L. Yakymiv, “Limit theorem for the general number of cycles in a random $A$-permutation”, Teor. Veroyatnost. i Primenen., 52:1 (2007), 69–83; Theory Probab. Appl., 52:1 (2008), 133–146

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, “On the Number of $A$-Mappings”, Math. Notes, 86:1 (2009), 132–139
2. A. L. Yakymiv, “Limit Theorem for the Middle Members of Ordered Cycle Lengths in Random $A$-Permutations”, Theory Probab. Appl., 54:1 (2010), 114–128
3. 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
4. A. L. Yakymiv, “Random $A$-permutations and Brownian motion”, Proc. Steklov Inst. Math., 282 (2013), 298–318
5. A. L. Yakymiv, “On a number of components in a random $A$-mapping”, Theory Probab. Appl., 59:1 (2015), 114–127
6. A. L. Yakymiv, “Tauberian theorem for generating functions of multiple series”, Theory Probab. Appl., 60:2 (2016), 343–347
7. A. L. Yakymiv, “A Tauberian theorem for multiple power series”, Sb. Math., 207:2 (2016), 286–313
8. Betz V., Schaefer H., “The Number of Cycles in Random Permutations Without Long Cycles Is Asymptotically Gaussian”, ALEA-Latin Am. J. Probab. Math. Stat., 14:1 (2017), 427–444
9. Betz V. Schaefer H. Zeindler D., “Random Permutations Without Macroscopic Cycles”, Ann. Appl. Probab., 30:3 (2020), 1484–1505
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