Abstract not found

We show that the Poisson–Dirichlet distribution is the distribution of points in a scaleinvariant 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 � 1. Restricting both processes to (0,β] for 0 <β � 1, we give an explicit formula for the total variation distance between their distributions. Connections between various representations of the Poisson–Dirichlet process are discussed.

We consider finite-state time-nonhomogeneous Markov chains whose transition matrix at time n is I + G/nζ where G is a “generator ” matrix, that is G(i,j)> 0 for i,j distinct, and G(i,i) = − ∑ k̸=i G(i,k), and ζ> 0 is a strength parameter. In these chains, as time grows, the positions are less and less likely to change, and so form simple models of age-dependent time-reinforcing schemes. These chains, however, exhibit some different, perhaps unexpected, occupation behaviors depending on parameters. Although it is shown, on the one hand, that the position at time n converges to a point-mixture for all ζ> 0, on the other hand, the average occupation vector up to time n, when variously 0 < ζ < 1, ζ> 1 or ζ = 1, is seen to converge to a constant, a pointmixture, or a distribution µG with no atoms and full support on a simplex respectively, as n ↑ ∞. This last type of limit can be interpreted as a sort of “spreading ” between the cases 0 < ζ < 1 and ζ> 1. In particular, when G is appropriately chosen, intriguingly, µG is a Dirichlet distribution, reminiscent of results in Pólya urns. Research supported in part by nsa-h982300510041 and NSF-DMS-0504193 Key words and phrases: laws of large numbers, nonhomogeneous, Markov, occupation, reinforcement, Dirichlet distribution.