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2006•Combinatorics Probability ComputingOpen access

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

Richard Arratia, A. D. Barbour, Simon Tavaré

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Abstract

For a random permutation of 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.

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For a random permutation of 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.

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Available abstract

For a random permutation of 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.

Key concepts: Mathematics, Poisson distribution, Dirichlet distribution, Lemma (botany), Mathematical proof, Random graph, Combinatorics, Permutation (music)

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