The Annals of Probability

The Cycle Structure of Random Permutations

Richard Arratia and Simon Tavare

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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.

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

First available in Project Euclid: 19 April 2007

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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

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


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

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