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15
The twoparameter PoissonDirichlet distribution derived from a stable subordinator.
, 1995
"... The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov ..."
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Cited by 305 (33 self)
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The twoparameter PoissonDirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual PoissonDirichlet distribution with a single parameter `, introduced by Kingman, is pd(0; `). Known properties of pd(0; `), including the Markov chain description due to VershikShmidtIgnatov, are generalized to the twoparameter case. The sizebiased random permutation of pd(ff; `) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For 0 ! ff ! 1, pd(ff; 0) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index ff. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950's and 60's. The distribution of ranked lengths of e...
Random Brownian Scaling Identities and Splicing of Bessel Processes
 ANN. PROBAB
, 1997
"... An identity in distribution due to F. Knight for Brownian motion is extended in two different ways: firstly by replacing the supremum of a reflecting Brownian motion by the range of an unreflected Brownian motion, and secondly by replacing the reflecting Brownian motion by a recurrent Bessel process ..."
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Cited by 9 (5 self)
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An identity in distribution due to F. Knight for Brownian motion is extended in two different ways: firstly by replacing the supremum of a reflecting Brownian motion by the range of an unreflected Brownian motion, and secondly by replacing the reflecting Brownian motion by a recurrent Bessel process. Both extensions are explained in terms of random Brownian scaling transformations and Brownian excursions. The first extension is related to two different constructions of Ito's law of Brownian excursions, due to D. Williams and J.M. Bismut, each involving backtoback splicing of fragments of two independent threedimensional Bessel processes. Generalizations of both splicing constructions are described which involve Bessel processes and Bessel bridges of arbitrary positive real dimension.
On the lengths of excursions of some Markov processes
 In S'eminaire de Probabilit'es XXXI
, 1996
"... Results are obtained regarding the distribution of the ranked lengths of component intervals in the complement of the random set of times when a recurrent Markov process returns to its starting point. Various martingales are described in terms of the L'evy measure of the Poisson point process o ..."
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Cited by 8 (5 self)
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Results are obtained regarding the distribution of the ranked lengths of component intervals in the complement of the random set of times when a recurrent Markov process returns to its starting point. Various martingales are described in terms of the L'evy measure of the Poisson point process of interval lengths on the local time scale. The martingales derived from the zero set of a onedimensional diffusion are related to martingales studied by Az'ema and Rainer. Formulae are obtained which show how the distribution of interval lengths is affected when the underlying process is subjected to a Girsanov transformation. In particular, results for the zero set of an OrnsteinUhlenbeck process or a CoxIngersollRoss process are derived from results for a Brownian motion or recurrent Bessel process, when the zero set is the range of a stable subordinator. 1 Introduction Let Z be the random set of times that a recurrent diffusion process X returns to its starting state 0. For a fixed or ra...
Gamma tilting calculus for GGC and Dirichlet means with applications to Linnik processes and occupation time laws for randomly skewed Bessel processes and bridges
, 2006
"... This paper develops some general calculus for GGC and Dirichlet process means functionals. It then proceeds via an investigation of positive Linnik random variables, and more generally random variables derived from compositions of a stable subordinator with GGC subordinators, to establish various di ..."
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Cited by 8 (6 self)
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This paper develops some general calculus for GGC and Dirichlet process means functionals. It then proceeds via an investigation of positive Linnik random variables, and more generally random variables derived from compositions of a stable subordinator with GGC subordinators, to establish various distributional equivalences between these models and phenomena connected to local times and occupation times of what are defined as randomly skewed Bessel processes and bridges. This yields a host of interesting identities and explicit density formula for these models. Randomly skewed Bessel processes and bridges may be seen as a randomization of their pskewed counterparts developed in Barlow, Pitman and Yor (1989) and Pitman and Yor (1997), and are shown to naturally arise via exponential tilting. As a special result it is shown that the occupation time of a pskewed random Bessel process or (generalized) bridge is equivalent in distribution to the the occupation time of a nontrivial randomly skewed process. 1
Distributions of linear functionals of the two parameter Poisson–Dirichlet random measures
 ANN. APPL. PROBAB
, 2008
"... The present paper provides exact expressions for the probability distributions of linear functionals of the twoparameter Poisson– Dirichlet process PD(α,θ). We obtain distributional results yielding exact forms for density functions of these functionals. Moreover, several interesting integral ident ..."
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Cited by 8 (5 self)
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The present paper provides exact expressions for the probability distributions of linear functionals of the twoparameter Poisson– Dirichlet process PD(α,θ). We obtain distributional results yielding exact forms for density functions of these functionals. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean of a Poisson–Dirichlet process and the mean of a suitable Dirichlet process. Finally, some distributional characterizations in terms of mixture representations are proved. The usefulness of the results contained in the paper is demonstrated by means of some illustrative examples. Indeed, our formulae are relevant to occupation time phenomena connected with Brownian motion and more general Bessel processes, as well as to models arising in Bayesian nonparametric statistics.
The Double CFTP method
, 2010
"... Abstract. We consider the problem of the exact simulation of random variables Z that satisfy the distributional identity Z L = V Y + (1 − V)Z, where V ∈ [0, 1] and Y are independent, and L = denotes equality in distribution. Equivalently, Z is the limit of a Markov chain driven by that map. We give ..."
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Cited by 6 (2 self)
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Abstract. We consider the problem of the exact simulation of random variables Z that satisfy the distributional identity Z L = V Y + (1 − V)Z, where V ∈ [0, 1] and Y are independent, and L = denotes equality in distribution. Equivalently, Z is the limit of a Markov chain driven by that map. We give an algorithm that can be automated under the condition that we have a source capable of generating independent copies of Y, and that V has a density that can be evaluated in a black box format. The method uses a doubling trick for inducing coalescence in coupling from the past. Applications include exact samplers for many Dirichlet means, some twoparameter Poisson–Dirichlet means, and a host of other distributions related to occupation times of Bessel bridges that can be described by stochastic fixed point equations. Keywords and phrases. Random variate generation. Perpetuities. Coupling from the past. Random partitions. Stochastic recurrences. Stochastic fixed point equations. Distribution theory. Markov chain Monte Carlo. Simulation. Expected time analysis. Bessel bridge. PoissonDirichlet. Dirichlet means.