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23
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.
Applications of the continuoustime ballot theorem to Brownian motion and related processes
, 2001
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Models for spatially distributed populations: The effect of withinpatch variability
, 1981
"... This paper studies population models which have the following three ingredients: populations are divided into local subpopulations, local population dynamics are noniinear and random events occur locally in space. In this setting local stochastic phenomena have a systematic effect on average populat ..."
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Cited by 8 (3 self)
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This paper studies population models which have the following three ingredients: populations are divided into local subpopulations, local population dynamics are noniinear and random events occur locally in space. In this setting local stochastic phenomena have a systematic effect on average population density and this effect does not disappear in large populations. This result is an outcome of the interaction of the three ingredients in the models and it says that stochastic models of systems of patches can be expected to give results for average population density that differ systematically from those of deterministic models. The magnitude of these differences is related to the degree of nonlinearity of local dynamics and the magnitude of local variability. These results explain those obtained from a number of previously published models which give conclusions that differ from those of deterministic models. Results are also obtained that show how stochastic models of systems of patches may be simplified to facilitate their study. 1. INTR~OUCTI~N The chances of survival and reproduction for an individual organism
More uses of exchangeability: Representations of complex random structures
 Probability and Mathematical Genetics: Papers in Honour of Sir
, 2010
"... We review old and new uses of exchangeability, emphasizing the general theme of exchangeable representations of complex random structures. Illustrations of this theme include processes of stochastic coalescence and fragmentation; continuum random trees; secondorder limits of distances in random gra ..."
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Cited by 8 (1 self)
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We review old and new uses of exchangeability, emphasizing the general theme of exchangeable representations of complex random structures. Illustrations of this theme include processes of stochastic coalescence and fragmentation; continuum random trees; secondorder limits of distances in random graphs; isometry classes of metric spaces with probability measures; limits of dense random graphs; and more sophisticated uses in finitary combinatorics.
Dual random fragmentation and coagulation and an application to the genealogy of Yule processes
 In Mathematics and Computer Science III (Vienna 2004), Trends Math
, 2004
"... The purpose of this work is to describe a duality between a fragmentation associated to certain Dirichlet distributions and a natural random coagulation. The dual fragmentation and coalescent chains arising in this setting appear in the description of the genealogy of Yule processes. 1 ..."
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Cited by 7 (2 self)
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The purpose of this work is to describe a duality between a fragmentation associated to certain Dirichlet distributions and a natural random coagulation. The dual fragmentation and coalescent chains arising in this setting appear in the description of the genealogy of Yule processes. 1
Limits of compound and thinned point processes
 J. Appl. Probab
, 1975
"... Let n = LOr be a point process on some space S and let j j 6,61,6 2, •.. be identically distributed nonnegative random variables which are mutually independent and independent of n. We can then form the compound point process ~ = L6 j O ..."
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Cited by 5 (1 self)
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Let n = LOr be a point process on some space S and let j j 6,61,6 2, •.. be identically distributed nonnegative random variables which are mutually independent and independent of n. We can then form the compound point process ~ = L6 j O
The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity
, 2004
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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 of int ..."
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Cited by 4 (3 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...
MARKOVIAN BRIDGES: WEAK CONTINUITY AND PATHWISE CONSTRUCTIONS
, 905
"... Abstract. A Markovian bridge is a probability measure taken from a disintegration of the law of an initial part of the path of a Markov process given its terminal value. As such, Markovian bridges admit a natural parameterization in terms of the state space of the process. In the context of Feller p ..."
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Cited by 4 (0 self)
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Abstract. A Markovian bridge is a probability measure taken from a disintegration of the law of an initial part of the path of a Markov process given its terminal value. As such, Markovian bridges admit a natural parameterization in terms of the state space of the process. In the context of Feller processes with continuous transition densities, we construct by weak convergence considerations the only versions of Markovian bridges which are weakly continuous with respect to their parameter. We use this weakly continuous construction to provide an extension of the strong Markov property in which the flow of time is reversed. In the context of selfsimilar Feller process, the last result is shown to be useful in the construction of Markovian bridges out of the trajectories of the original process.