Poisson-Dirichlet And Gem Invariant Distributions For Split-And-Merge Transformations Of An Interval Partition

The Poisson-Dirichlet law is the unique invariant distribution for uniform Split-Merge Transformations

Entropy of infinite systems and transformations

The Kolmogorov-Sinai entropy is a far reaching dynamical generalization of Shannon entropy of information systems. This entropy works perfectly for probability measure preserving (p.m.p.) transformations. However, it is not useful when there is no finite invariant measure. There are certain successful extensions of the notion of entropy to infinite measure spaces, or transformations with ...

Two-Parameter Poisson-Dirichlet Measures and Reversible Exchangeable Fragmentation-Coalescence Processes

We show that for 0 < α < 1 and θ > −α, the Poisson-Dirichlet distribution with parameter (α, θ) is the unique reversible distribution of a rather natural fragmentation-coalescence process. This completes earlier results in the literature for certain split and merge transformations and the parameter α = 0. DOI: https://doi.org/10.1017/S0963548307008784 Posted at the Zurich Open Repository and Ar...

The Poisson-Dirichlet Distribution And The Scale-Invariant Poisson Process

We show that the Poisson–Dirichlet distribution is the distribution of points in a scale-invariant Poisson process, conditioned on the event that the sum T of the locations of the points in (0,1] is 1. This extends to a similar result, rescaling the locations by T , and conditioning on the event that T 6 1. Restricting both processes to (0, β] for 0 < β 6 1, we give an explicit formula for the ...

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

For a random permutation of n objects, as n → ∞, 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-Diri...