The Annals of Probability

The Cycle Structure of Random Permutations

Richard Arratia and Simon Tavare

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Abstract

The total variation distance between the process which counts cycles of size $1,2,\ldots, b$ of a random permutation of $n$ objects and a process $(Z_1,Z_2,\ldots, Z_b)$ of independent Poisson random variables with $\mathbb{E}Z_i = 1/i$ converges to 0 if and only if $b/n \rightarrow 0$. This Poisson approximation can be used to give simple proofs of limit theorems and bounds for a wide variety of functionals of random permutations. These limit theorems include the Erdos-Turan theorem for the asymptotic normality of the log of the order of a random permutation, and the DeLaurentis-Pittel functional central limit theorem for the cycle sizes. We give a simple explicit upper bound on the total variation distance to show that this distance decays to zero superexponentially fast as a function of $n/b \rightarrow \infty$. A similar result holds for derangements and, more generally, for permutations conditioned to have given numbers of cycles of various sizes. Comparison results are included to show that in approximating the cycle structure by an independent Poisson process the main discrepancy arises from independence rather than from Poisson marginals.

Article information

Source
Ann. Probab. Volume 20, Number 3 (1992), 1567-1591.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176989707

Digital Object Identifier
doi:10.1214/aop/1176989707

Mathematical Reviews number (MathSciNet)
MR1175278

Zentralblatt MATH identifier
0759.60007

JSTOR
links.jstor.org

Subjects
Primary: 60C05: Combinatorial probability
Secondary: 60F17: Functional limit theorems; invariance principles 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 60G18: Self-similar processes 05A05: Permutations, words, matrices 05A16: Asymptotic enumeration

Keywords
Poisson process inclusion-exclusion total variation exponential generating functions derangements conditional limit theorems

Citation

Arratia, Richard; Tavare, Simon. The Cycle Structure of Random Permutations. Ann. Probab. 20 (1992), no. 3, 1567--1591. doi:10.1214/aop/1176989707. https://projecteuclid.org/euclid.aop/1176989707


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