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 Diskr. Mat., 2010, Volume 22, Issue 1, Pages 126–149 (Mi dm1089)

A limit theorem for the logarithm of the order of a random $A$-permutation

A. L. Yakymiv

Abstract: In this article, a random permutation $\tau_n$ is considered which is uniformly distributed on the set of all permutations of degree $n$ whose cycle lengths lie in a fixed set $A$ (the so-called $A$-permutations). It is assumed that the set $A$ has an asymptotic density $\sigma>0$, and $|k\colon k\leq n, k\in A, m-k\in A|/n\to\sigma^2$ as $n\to\infty$ uniformly in $m\in[n,Cn]$ for an arbitrary constant $C>1$. The minimum degree of a permutation such that it becomes equal to the identity permutation is called the order of permutation. Let $Z_n$ be the order of a random permutation $\tau_n$. In this article, it is shown that the random variable $\ln Z_n$ is asymptotically normal with mean $l(n)=\sum_{k\in A(n)}\ln(k)/k$ and variance $\sigma\ln^3(n)/3$, where $A(n)=\{k\colon k\in A, k\leq n\}$. This result generalises the well-known theorem of P. Erdős and P. Turán where the uniform distribution on the whole symmetric group of permutations $S_n$ is considered, i.e., where $A$ is equal to the set of positive integers $\mathbb N$.

DOI: https://doi.org/10.4213/dm1089

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English version:
Discrete Mathematics and Applications, 2010, 20:3, 247–275

Bibliographic databases:

UDC: 519.2

Citation: A. L. Yakymiv, “A limit theorem for the logarithm of the order of a random $A$-permutation”, Diskr. Mat., 22:1 (2010), 126–149; Discrete Math. Appl., 20:3 (2010), 247–275

Citation in format AMSBIB
\Bibitem{Yak10} \by A.~L.~Yakymiv \paper A limit theorem for the logarithm of the order of a~random $A$-permutation \jour Diskr. Mat. \yr 2010 \vol 22 \issue 1 \pages 126--149 \mathnet{http://mi.mathnet.ru/dm1089} \crossref{https://doi.org/10.4213/dm1089} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2676236} \zmath{https://zbmath.org/?q=an:05773259} \elib{https://elibrary.ru/item.asp?id=20730329} \transl \jour Discrete Math. Appl. \yr 2010 \vol 20 \issue 3 \pages 247--275 \crossref{https://doi.org/10.1515/DMA.2010.015} \elib{https://elibrary.ru/item.asp?id=22058818} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77954729843} 

• http://mi.mathnet.ru/eng/dm1089
• https://doi.org/10.4213/dm1089
• http://mi.mathnet.ru/eng/dm/v22/i1/p126

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Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. G. I. Ivchenko, M. V. Soboleva, “Some nonequiprobable models of random permutations”, Discrete Math. Appl., 21:4 (2011), 397–406
2. A. L. Yakymiv, “A Generalization of the Curtiss Theorem for Moment Generating Functions”, Math. Notes, 90:6 (2011), 920–924
3. A. L. Yakymiv, “Random $A$-permutations and Brownian motion”, Proc. Steklov Inst. Math., 282 (2013), 298–318
4. A. L. Yakymiv, “On a number of components in a random $A$-mapping”, Theory Probab. Appl., 59:1 (2015), 114–127
5. A. L. Yakymiv, “Tauberian theorem for generating functions of multiple series”, Theory Probab. Appl., 60:2 (2016), 343–347
6. Storm J., Zeindler D., “the Order of Large Random Permutations With Cycle Weights”, Electron. J. Probab., 20 (2015), 126, 1–34
7. A. L. Yakymiv, “A Tauberian theorem for multiple power series”, Sb. Math., 207:2 (2016), 286–313
8. Storm J., Zeindler D., “Total variation distance and the Erdős–Turán law for random permutations with polynomially growing cycle weights”, Ann. Inst. Henri Poincare-Probab. Stat., 52:4 (2016), 1614–1640
9. A. L. Yakymiv, “Limit theorems for the logarithm of the order of a random $A$-mapping”, Discrete Math. Appl., 27:5 (2017), 325–338
10. A. L. Yakymiv, “On the order of random permutation with cycle weights”, Theory Probab. Appl., 63:2 (2018), 209–226
11. A. L. Yakymiv, “Asymptotics with remainder term for moments of the total cycle number of random $A$-permutation”, Discrete Math. Appl., 31:1 (2021), 51–60
12. A. L. Yakymiv, “Size distribution of the largest component of a random $A$-mapping”, Discrete Math. Appl., 31:2 (2021), 145–153
13. A. L. Yakymiv, “Abelian theorem for the regularly varying measure and its density in orthant”, Theory Probab. Appl., 64:3 (2019), 385–400
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