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July, 1992 The Cycle Structure of Random Permutations
Richard Arratia, Simon Tavare
Ann. Probab. 20(3): 1567-1591 (July, 1992). DOI: 10.1214/aop/1176989707

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.

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Richard Arratia. Simon Tavare. "The Cycle Structure of Random Permutations." Ann. Probab. 20 (3) 1567 - 1591, July, 1992. https://doi.org/10.1214/aop/1176989707

Information

Published: July, 1992
First available in Project Euclid: 19 April 2007

zbMATH: 0759.60007
MathSciNet: MR1175278
Digital Object Identifier: 10.1214/aop/1176989707

Subjects:
Primary: 60C05
Secondary: 05A05 , 05A16 , 60B15 , 60F17 , 60G18

Keywords: conditional limit theorems , derangements , exponential generating functions , inclusion-exclusion , Poisson process , Total variation

Rights: Copyright © 1992 Institute of Mathematical Statistics

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Vol.20 • No. 3 • July, 1992
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