# A tale of three couplings: Poisson-Dirichlet and GEM approximations for random permutations

Arratia, R; Barbour, A D; Tavaré, S (2006). A tale of three couplings: Poisson-Dirichlet and GEM approximations for random permutations. Combinatorics, Probability & Computing, 15(1-2):31-62.

## Abstract

For a random permutation of $n$ objects, as $n \to \infty$, the process giving the proportion of elements in the longest cycle, the second-longest cycle, and so on, converges in distribution to the Poisson–Dirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmidt. For soft reasons, this is equivalent to the statement that the random permutations and the Poisson–Dirichlet process can be coupled so that zero is the limit of the expected $\ell_1$ distance between the process of cycle length proportions and the Poisson–Dirichlet process. We investigate how rapid this metric convergence can be, and in doing so, give two new proofs of the distributional convergence.

One of the couplings we consider has an analogue for the prime factorizations of a uniformly distributed random integer, and these couplings rely on the ‘scale-invariant spacing lemma’ for the scale-invariant Poisson processes, proved in this paper.

For a random permutation of $n$ objects, as $n \to \infty$, the process giving the proportion of elements in the longest cycle, the second-longest cycle, and so on, converges in distribution to the Poisson–Dirichlet process with parameter 1. This was proved in 1977 by Kingman and by Vershik and Schmidt. For soft reasons, this is equivalent to the statement that the random permutations and the Poisson–Dirichlet process can be coupled so that zero is the limit of the expected $\ell_1$ distance between the process of cycle length proportions and the Poisson–Dirichlet process. We investigate how rapid this metric convergence can be, and in doing so, give two new proofs of the distributional convergence.

One of the couplings we consider has an analogue for the prime factorizations of a uniformly distributed random integer, and these couplings rely on the ‘scale-invariant spacing lemma’ for the scale-invariant Poisson processes, proved in this paper.

## Citations

4 citations in Web of Science®
3 citations in Scopus®

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Item Type: Journal Article, refereed, original work 07 Faculty of Science > Institute of Mathematics 510 Mathematics English 2006 05 Jan 2010 15:00 05 Apr 2016 13:23 Cambridge University Press 0963-5483 Copyright © 2006 Cambridge University Press 10.1017/S0963548305007054
Permanent URL: http://doi.org/10.5167/uzh-21590