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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 ..."
Abstract

Cited by 218 (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.
Two coalescents derived from the ranges of stable subordinators
 Electron. J. Probab
"... E l e c t r o n ..."
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 7 (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
, 906
"... This note is an announcement of the results. The full version of this paper, including full proofs, will be available soon. 1. ..."
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This note is an announcement of the results. The full version of this paper, including full proofs, will be available soon. 1.
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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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.
Hitting halfspaces by BesselBrownian di usions
, 2009
"... The purpose of the paper is to nd explicit formulas describing the joint distributions of the rst hitting time and place for halfspaces of codimension one for a di usion in Rn+1, composed of onedimensional Bessel process and independent ndimensional Brownian motion. The most important argument is ..."
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The purpose of the paper is to nd explicit formulas describing the joint distributions of the rst hitting time and place for halfspaces of codimension one for a di usion in Rn+1, composed of onedimensional Bessel process and independent ndimensional Brownian motion. The most important argument is carried out for the twodimensional situation. We show that this amounts to computation of distributions of various integral functionals with respect to a twodimensional process with independent Bessel components. As a result, we provide a formula for the Poisson kernel of a halfspace or of a strip for the operator (I − ∆) α/2, 0 < α < 2. In the case of a halfspace, this result was recently found, by di erent methods, in [6]. As an application of our method we also compute various formulas for rst hitting places for the isotropic stable Lévy process. 1
Regularising Mappings of Lévy Measures ISSN 13982699Regularising Mappings of Lévy Measures
, 2004
"... In this paper we introduce and study a regularising onetoone mapping Υ0 from the class of onedimensional Lévy measures into itself. This mapping appeared implicitly in our previous paper [BT2], where we introduced a onetoone mapping Υ from the class ID(∗) of onedimensional infinitely divisible ..."
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In this paper we introduce and study a regularising onetoone mapping Υ0 from the class of onedimensional Lévy measures into itself. This mapping appeared implicitly in our previous paper [BT2], where we introduced a onetoone mapping Υ from the class ID(∗) of onedimensional infinitely divisible probability measures into itself. Based on the studies of Υ0 in the present paper, we also deduce further properties of Υ. In particular it is proved that Υ maps the class L(∗) of selfdecomposable laws onto the so called Thorin class T(∗). Finally, partly motivated by our previous studies of infinite divisibility in free probability, we introduce a oneparameter family (Υ α) α∈[0,1] of onetoone mappings Υ α: ID(∗) → ID(∗), which interpolates smoothly between Υ (α = 0) and the identity mapping on ID(∗) (α = 1). We prove that each of the mappings Υ α shares the properties of Υ exhibited in [BT2]. In particular, they are representable in terms of stochastic integrals with respect to associated Levy processes.