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22
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 364 (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...
Harmonic functions on multiplicative graphs and interpolation polynomials, Electron
 J. Combin. 7 (2000), Research paper
"... Abstract. We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur’ ..."
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Cited by 26 (12 self)
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Abstract. We construct examples of nonnegative harmonic functions on certain graded graphs: the Young lattice and its generalizations. Such functions first emerged in harmonic analysis on the infinite symmetric group. Our method relies on multivariate interpolation polynomials associated with Schur’s S and P functions and with Jack symmetric functions. As a by–product, we compute certain Selberg–type integrals.
A twoparameter family infinitedimensional diffusions in the Kingman simplex
, 2007
"... The main result of the present paper is to construct a twoparameter family of Markov processes Xα,θ(t) in the infinitedimensional Kingman simplex ..."
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Cited by 16 (4 self)
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The main result of the present paper is to construct a twoparameter family of Markov processes Xα,θ(t) in the infinitedimensional Kingman simplex
Coagulationfragmentation duality, Poisson–Dirichlet distributions and random recursive trees
 Ann. Appl. Probab
, 2006
"... In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the twoparameter family of Poisson–Dirichlet distributions PD(α, θ) that take values in ..."
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Cited by 16 (1 self)
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In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the twoparameter family of Poisson–Dirichlet distributions PD(α, θ) that take values in this space. We introduce families of random fragmentation and coagulation operators Frag α and Coag α,θ, respectively, with the following property: if the input to Frag α has PD(α, θ) distribution, then the output has PD(α, θ +1) distribution, while the reverse is true for Coag α,θ. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between PD(α, θ) and PD(αβ, θ). Repeated application of the Frag α operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation–fragmentation duality. 1. Introduction. The
Random Discrete Distributions Derived From SelfSimilar Random Sets
 Electronic J. Probability
, 1996
"... : A model is proposed for a decreasing sequence of random variables (V 1 ; V 2 ; \Delta \Delta \Delta) with P n V n = 1, which generalizes the PoissonDirichlet distribution and the distribution of ranked lengths of excursions of a Brownian motion or recurrent Bessel process. Let V n be the length ..."
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Cited by 15 (7 self)
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: A model is proposed for a decreasing sequence of random variables (V 1 ; V 2 ; \Delta \Delta \Delta) with P n V n = 1, which generalizes the PoissonDirichlet distribution and the distribution of ranked lengths of excursions of a Brownian motion or recurrent Bessel process. Let V n be the length of the nth longest component interval of [0; 1]nZ, where Z is an a.s. nonempty random closed of (0; 1) of Lebesgue measure 0, and Z is selfsimilar, i.e. cZ has the same distribution as Z for every c ? 0. Then for 0 a ! b 1 the expected number of n's such that V n 2 (a; b) equals R b a v \Gamma1 F (dv) where the structural distribution F is identical to the distribution of 1 \Gamma sup(Z " [0; 1]). Then F (dv) = f(v)dv where (1 \Gamma v)f(v) is a decreasing function of v, and every such probability distribution F on [0; 1] can arise from this construction. Keywords: interval partition, zero set, excursion lengths, regenerative set, structural distribution. AMS subject classificat...
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 11 (10 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.
The MarkovKrein correspondence in several dimensions
, 1998
"... We compute the joint moments of several linear functionals with respect to a Dirichlet random measure and some of its generalizations. Given a probability distribution on a space X, let M = M denote the random probability measure on X known as Dirichlet random measure with the parameter distributi ..."
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Cited by 8 (0 self)
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We compute the joint moments of several linear functionals with respect to a Dirichlet random measure and some of its generalizations. Given a probability distribution on a space X, let M = M denote the random probability measure on X known as Dirichlet random measure with the parameter distribution . We prove the formula D 1 1 \Gamma z 1 F 1 (M) \Gamma : : : \Gamma z mFm (M) E = exp Z ln 1 1 \Gamma z 1 f 1 (x) \Gamma : : : \Gamma z m fm (x) (dx); where F k (M) = R X f k (x)M(dx), the angle brackets denote the average in M, and f 1 ; : : : ; fm are the coordinates of a map f : X ! R m . The formula describes implicitly the joint distribution of the random variables F k (M), k = 1; : : : ; m. Assuming that the joint moments p k1 ;::: ;k m = R f k1 1 (x) : : : f km m (x)d (x) are all finite, we restate the above formula as an explicit description of the joint moments of the variables F 1 ; : : : ; Fm in terms of p k1 ;::: ;k m . In case of a finite space, jXj = N +...
AN ASYMPTOTIC SAMPLING FORMULA FOR THE COALESCENT WITH RECOMBINATION
"... Ewens sampling formula (ESF) is a oneparameter family of probability distributions with a number of intriguing combinatorial connections. This elegant closedform formula first arose in biology as the stationary probability distribution of a sample configuration at one locus under the infinitealle ..."
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Cited by 7 (0 self)
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Ewens sampling formula (ESF) is a oneparameter family of probability distributions with a number of intriguing combinatorial connections. This elegant closedform formula first arose in biology as the stationary probability distribution of a sample configuration at one locus under the infinitealleles model of mutation. Since its discovery in the early 1970s, the ESF has been used in various biological applications, and has sparked several interesting mathematical generalizations. In the population genetics community, extending the underlying randommating model to include recombination has received much attention in the past, but no general closedform sampling formula is currently known even for the simplest extension, that is, a model with two loci. In this paper, we show that it is possible to obtain useful closedform results in the case the populationscaled recombination rate ρ is large but not necessarily infinite. Specifically, we consider an asymptotic expansion of the twolocus sampling formula in inverse powers of ρ and obtain closedform expressions for the first few terms in the expansion. Our asymptotic sampling formula applies to arbitrary sample sizes and configurations. 1. Introduction. The
Twoparameter family of diffusion processes in the Kingman simplex, arXiv: math.PR/0708.1930
, 2007
"... §1. The Kingman graph §2. Symmetric algebra §3. Symmetric functions and the Kingman simplex ..."
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§1. The Kingman graph §2. Symmetric algebra §3. Symmetric functions and the Kingman simplex
Random walks on strict partitions
"... We consider a certain sequence of random walks. The state space of the nth random walk is the set of all strict partitions of n (that is, partitions without equal parts). We prove that, as n goes to infinity, these random walks converge to a continuoustime Markov process. The state space of this pr ..."
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Cited by 3 (2 self)
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We consider a certain sequence of random walks. The state space of the nth random walk is the set of all strict partitions of n (that is, partitions without equal parts). We prove that, as n goes to infinity, these random walks converge to a continuoustime Markov process. The state space of this process is the infinitedimensional simplex consisting of all nonincreasing infinite sequences of nonnegative numbers with sum less than or equal to one. The main result about the limit process is the expression of its the pregenerator as a formal second order differential operator in a polynomial algebra. Of separate interest is the generalization of Kerov interlacing coordinates to the case of shifted Young diagrams. 1