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

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 proport Show more