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The PoissonDirichlet distribution and the scaleinvariant Poisson process
 COMBIN. PROBAB. COMPUT
, 1999
"... 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 ..."
Abstract

Cited by 8 (1 self)
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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.
Occupation laws for some timenonhomogeneous Markov chains
, 2008
"... We consider finitestate timenonhomogeneous 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 ..."
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We consider finitestate timenonhomogeneous 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 agedependent timereinforcing 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 pointmixture 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 nsah982300510041 and NSFDMS0504193 Key words and phrases: laws of large numbers, nonhomogeneous, Markov, occupation, reinforcement, Dirichlet distribution.
Competition between Discrete Random Variables, with Applications to Occupancy Problems
, 2008
"... Consider n players whose “scores ” are independent and identically distributed values {Xi} n i=1 from some discrete distribution F. We pay special attention to the cases where (i) F is geometric with parameter p → 0 and (ii) F is uniform on {1,2,...,N}; the latter case clearly corresponds to the cla ..."
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Consider n players whose “scores ” are independent and identically distributed values {Xi} n i=1 from some discrete distribution F. We pay special attention to the cases where (i) F is geometric with parameter p → 0 and (ii) F is uniform on {1,2,...,N}; the latter case clearly corresponds to the classical occupancy problem. The quantities of interest to us are, first, the Ustatistic W which counts the number of “ties” between pairs i,j; second, the univariate statistic Yr, which counts the number of strict rway ties between contestants, i.e., episodes of the 1 form Xi1 = Xi2 =... = Xir; Xj = Xi1;j = i1,i2,...,ir; and, last but not least, the multivariate vector ZAB = (YA,YA+1,...,YB). We provide Poisson approximations for the distributions of W, Yr and ZAB under some general conditions. New results on the joint distribution of cell counts in the occupancy problem are derived as a corollary. 1