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24
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 347 (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...
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 extende ..."
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Cited by 47 (24 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.
Two coalescents derived from the ranges of stable subordinators
 Electron. J. Probab
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Spectral analysis of subordinate Brownian motions in halfline. Preliminary version
 Preprint, 2010, arXiv:1006.0524v1. SUBORDINATE BROWNIAN MOTIONS IN HALFLINE 57
"... Abstract. We study onedimensional Lévy processes with LévyKhintchine exponent ψ(ξ2), where ψ is a complete Bernstein function. These processes are subordinate Brownian motions corresponding to subordinators, whose Lévy measure has completely monotone density; or, equivalently, symmetric Lévy proce ..."
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Cited by 14 (2 self)
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Abstract. We study onedimensional Lévy processes with LévyKhintchine exponent ψ(ξ2), where ψ is a complete Bernstein function. These processes are subordinate Brownian motions corresponding to subordinators, whose Lévy measure has completely monotone density; or, equivalently, symmetric Lévy processes whose Lévy measure has completely monotone density on (0,∞). Examples include symmetric stable processes and relativistic processes. The main result is a formula for the generalized eigenfunctions of transition operators of the process killed after exiting the halfline. A generalized eigenfunction expansion of the transition operators is derived. As an application, a formula for the distribution of the first passage time (or the supremum functional) is obtained.
Applications of the continuoustime ballot theorem to Brownian motion and related processes
, 2001
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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.
SPECTRAL PROPERTIES OF THE CAUCHY PROCESS ON HALFLINE AND INTERVAL
, 906
"... Abstract. We study the spectral properties of the transition semigroup of the killed onedimensional Cauchy process on the halfline (0, ∞) and the interval (−1, 1). This process is related to the square root of onedimensional Laplacian A = − − d2 dx2 with a Dirichlet exterior condition (on a comp ..."
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Cited by 9 (2 self)
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Abstract. We study the spectral properties of the transition semigroup of the killed onedimensional Cauchy process on the halfline (0, ∞) and the interval (−1, 1). This process is related to the square root of onedimensional Laplacian A = − − d2 dx2 with a Dirichlet exterior condition (on a complement of a domain), and to a mixed Steklov problem in the halfplane. For the halfline, an explicit formula for generalized eigenfunctions ψλ of A is derived, and then used to construct spectral representation of A. Explicit formulas for the transition density of the killed Cauchy process in the halfline (or the heat kernel of A in (0, ∞)), and for the distribution of the first exit time from the halfline follow. The formula for ψλ is also used to construct approximations to eigenfunctions of A in the interval. For the eigenvalues λn of A in the interval the asymptotic formula λn = nπ π 1 − + O ( ) is 2 8 n derived, and all eigenvalues λn are proved to be simple. Finally, efficient numerical methods of estimation of eigenvalues λn are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to 9th decimal point. 1.
Some explicit Krein representations of certain subordinators, including the Gamma process
, 2005
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On Neumann Type Problems for Nonlocal Equations Set in a Half Space
"... Abstract. We study Neumann type boundary value problems for nonlocal equations related to Lévy processes. Since these equations are nonlocal, Neumann type problems can be obtained in many ways, depending on the kind of “reflection ” we impose on the outside jumps. To focus on the new phenomenas a ..."
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Cited by 1 (0 self)
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Abstract. We study Neumann type boundary value problems for nonlocal equations related to Lévy processes. Since these equations are nonlocal, Neumann type problems can be obtained in many ways, depending on the kind of “reflection ” we impose on the outside jumps. To focus on the new phenomenas and ideas, we consider different models of reflection and rather general nonsymmetric Lévy measures, but only simple linear equations in halfspace domains. We derive the Neumann/reflection problems through a truncation procedure on the Lévy measure, and then we develop a viscosity solution theory which includes comparison, existence, and some regularity results. For problems involving fractional Laplacian type operators like e.g. (−∆)α/2, we prove that solutions of all our nonlocal Neumann problems converge as α → 2 − to the solution of a classical Neumann problem. The reflection models we consider include cases where the underlying Lévy processes are reflected, projected, and/or censored upon exiting the domain. 1.