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 Mat. Zametki, 2007, Volume 81, Issue 6, Pages 939–947 (Mi mz3744)

Random $A$-Permutations: Convergence to a Poisson Process

A. L. Yakymiv

Steklov Mathematical Institute, Russian Academy of Sciences

Abstract: Suppose that $S_n$ is the permutation group of degree $n$, $A$ is a subset of the set of natural numbers $\mathbb N$, and $T_n=T_n(A)$ is the set of all permutations from $S_n$ whose cycle lengths belong to the set $A$. Permutations from $T_n$ are usually called $A$-permutations. We consider a wide class of sets $A$ of positive asymptotic density. Suppose that $\zeta_{mn}$ is the number of cycles of length $m$ of a random permutation uniformly distributed on $T_n$. It is shown in this paper that the finite-dimensional distributions of the random process $\{\zeta_{mn},m\in A\}$ weakly converge as $n\to\infty$ to the finite-dimensional distributions of a Poisson process on $A$.

Keywords: random permutation, Poisson process, permutation group, permutation cycle, total variance distance, normal distribution

DOI: https://doi.org/10.4213/mzm3744

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English version:
Mathematical Notes, 2007, 81:6, 840–846

Bibliographic databases:

UDC: 519.2
Revised: 19.09.2006

Citation: A. L. Yakymiv, “Random $A$-Permutations: Convergence to a Poisson Process”, Mat. Zametki, 81:6 (2007), 939–947; Math. Notes, 81:6 (2007), 840–846

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/mz3744
• https://doi.org/10.4213/mzm3744
• http://mi.mathnet.ru/eng/mz/v81/i6/p939

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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, “Limit Theorem for the Middle Members of Ordered Cycle Lengths in Random $A$-Permutations”, Theory Probab. Appl., 54:1 (2010), 114–128
2. 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
3. Benaych-Georges F., “Cycles of Free Words in Several Independent Random Permutations with Restricted Cycle Lengths”, Indiana Univ. Math. J., 59:5 (2010), 1547–1586
4. 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
5. Elboim D., Peled R., “Limit Distributions For Euclidean Random Permutations”, Commun. Math. Phys., 369:2 (2019), 457–522
6. Betz V., Schaefer H., Zeindler D., “Random Permutations Without Macroscopic Cycles”, Ann. Appl. Probab., 30:3 (2020), 1484–1505
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