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27
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 221 (37 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...
Arcsine laws and interval partitions derived from a stable subordinator
 Proc. London Math. Soc
, 1992
"... Le"vy discovered that the fraction of time a standard onedimensional Brownian motion B spends positive before time t has arcsine distribution, both for / a fixed time when B, #0 almost surely, and for / an inverse local time, when B, = 0 almost surely. This identity in distribution is extended fro ..."
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Cited by 44 (25 self)
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Le"vy discovered that the fraction of time a standard onedimensional Brownian motion B spends positive before time t has arcsine distribution, both for / a fixed time when B, #0 almost surely, and for / an inverse local time, when B, = 0 almost surely. This identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of B away from 0. Similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator, for instance a Bessel process of dimension d for 1.
Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models
 Ann. Probab
, 2008
"... Given any regularly varying dislocation measure, we identify a natural selfsimilar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous’s betasplitting models and Ford’s alpha models for phyl ..."
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Cited by 25 (10 self)
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Given any regularly varying dislocation measure, we identify a natural selfsimilar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous’s betasplitting models and Ford’s alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf.
On the Relative Lengths of Excursions Derived From a Stable Subordinator
, 1996
"... Results are obtained concerning the distribution of ranked relative lengths of excursions of a recurrent Markov process from a point in its state space whose inverse local time process is a stable subordinator. It is shown that for a large class of random times T the distribution of relative excursi ..."
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Cited by 14 (6 self)
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Results are obtained concerning the distribution of ranked relative lengths of excursions of a recurrent Markov process from a point in its state space whose inverse local time process is a stable subordinator. It is shown that for a large class of random times T the distribution of relative excursion lengths prior to T is the same as if T were a fixed time. It follows that the generalized arcsine laws of Lamperti extend to such random times T . For some other random times T , absolute continuity relations are obtained which relate the law of the relative lengths at time T to the law at a fixed time. 1 Introduction Following Lamperti [10], Wendel [24], Kingman [7], Knight [8], PermanPitman Yor [12, 13, 15], consider the sequence V 1 (T ) V 2 (T ) \Delta \Delta \Delta (1) of ranked lengths of component intervals of the set [0; T ]nZ, where T is a strictly positive random time, and Z is the zero set of a Markov process X started at zero, such as a Brownian motion or Bessel process,...
LIMITING LAWS ASSOCIATED WITH BROWNIAN MOTION PERTURBED BY NORMALIZED EXPONENTIAL WEIGHTS, I
, 2008
"... Let (Bt; t ≥ 0) be a one dimensional Brownian motion, with local time process (Lx t; t ≥ 0, x ∈ R). We determine the rate of decay of Z V [ { t (x): = Ex exp − 1 L ..."
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Cited by 14 (6 self)
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Let (Bt; t ≥ 0) be a one dimensional Brownian motion, with local time process (Lx t; t ≥ 0, x ∈ R). We determine the rate of decay of Z V [ { t (x): = Ex exp − 1 L
Perturbed Brownian Motions
 Probab. Th. Rel. Fields
, 1996
"... The paper deals with onedimensional Brownian motion perturbed when it hits its minimum and/or its maximum. It first presents some features of perturbed reflected Brownian motion defined as jBj \Gamma ¯` where B is standard Brownian motion, ` its local time at 0 and ¯ a positive constant; in partic ..."
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Cited by 12 (0 self)
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The paper deals with onedimensional Brownian motion perturbed when it hits its minimum and/or its maximum. It first presents some features of perturbed reflected Brownian motion defined as jBj \Gamma ¯` where B is standard Brownian motion, ` its local time at 0 and ¯ a positive constant; in particular it is shown that the positive and negative excursions in the sense of Ito of this "onesided" perturbed Brownian motion are two independent point processes, which in turn is used to construct twosided perturbed Brownian motion. It is then shown that the constructed process is almost surely the unique solution of the implicit stochastic equation studied by Le Gall [11], Carmona, Petit and Yor [5] and Davis [6], even when no restrictions are imposed on the partial reflections. Finally, the Hausdorff dimension of sets of exceptional points for perturbed Brownian motion such as points of monotonicity are computed. Mathematics Subject Classification (1991): 60J65 1 Introduction Over the ...
On the distribution of ranked heights of excursions of a Brownian bridge
 In preparation
, 1999
"... The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B br t ; 0 t 1) is described. The height M br+ j of the jth highest maximum over a positive excursion of the bridge has the same distribution as M br+ 1 =j, where th ..."
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Cited by 11 (6 self)
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The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B br t ; 0 t 1) is described. The height M br+ j of the jth highest maximum over a positive excursion of the bridge has the same distribution as M br+ 1 =j, where the distribution of M br+ 1 = sup 0t1 B br t is given by L'evy's formula P (M br+ 1 ? x) = e \Gamma2x 2 . The probability density of the height M br j of the jth highest maximum of excursions of the reflecting Brownian bridge (jB br t j; 0 t 1) is given by a modification of the known `function series for the density of M br 1 = sup 0t1 jB br t j. These results are obtained from a more general description of the distribution of ranked values of a homogeneous functional of excursions of the standardized bridge of a selfsimilar recurrent Markov process. Keywords: Brownian bridge, Brownian excursion, Brownian scaling, local time, selfsimilar recurrent Markov process, Bessel p...
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.
Distributions of functionals of the two parameter PoissonDirichlet process
, 2006
"... The present paper provides exact expressions for the probability distribution of linear functionals of the two–parameter Poisson–Dirichlet process PD(α, θ). Distributional results that follow from the application of an inversion formula for a (generalized) Cauchy– Stieltjes transform are achieved. ..."
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Cited by 7 (6 self)
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The present paper provides exact expressions for the probability distribution of linear functionals of the two–parameter Poisson–Dirichlet process PD(α, θ). Distributional results that follow from the application of an inversion formula for a (generalized) Cauchy– Stieltjes transform are achieved. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean functional of a Poisson–Dirichlet process and the mean functional of a suitable Dirichlet process. Finally, some distributional characterizations in terms of mixture representations are illustrated. 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.