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A Tale of Three Couplings: Poisson–Dirichlet and GEM Approximations for Random Permutations

Published online by Cambridge University Press:  03 January 2006

RICHARD ARRATIA ,
A. D. BARBOUR  and
SIMON TAVARÉ
RICHARD ARRATIA
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA (e-mail: rarratia@math.usc.edu)
A. D. BARBOUR
Affiliation:
Angewandte Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057, Zürich, Switzerland (e-mail: a.d.barbour@math.unizh.ch)
SIMON TAVARÉ
Affiliation:
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA and Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge, UK (e-mail: stavare@usc.edu)
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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.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 15 , Issue 1-2 , January 2006 , pp. 31 - 62
Copyright
2006 Cambridge University Press

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