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Permanent URL to this publication: http://dx.doi.org/10.5167/uzh-21590

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.

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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.

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Mathematics
DDC:510 Mathematics
Language:English
Date:2006
Deposited On:05 Jan 2010 16:00
Last Modified:27 Nov 2013 21:30
Publisher:Cambridge University Press
ISSN:0963-5483
Additional Information:Copyright © 2006 Cambridge University Press
Publisher DOI:10.1017/S0963548305007054
Citations:Web of Science®. Times Cited: 3
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