Showing papers on "Subordinator published in 1997"
TL;DR: The two-parameter Poisson-Dirichlet distribution with a single parameter is known as the size-biased random permutation (SBNP) as discussed by the authors, which was introduced by Engen in the context of species diversity and rediscovered by Perman and the authors in the study of excursions of Bessel processes.
Abstract: The two-parameter Poisson-Dirichlet distribution, denoted $\mathsf{PD}(\alpha, \theta)$ is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single parameter $\theta$, introduced by Kingman, is $\mathsf{PD}(0, \theta)$. Known properties of $\mathsf{PD}(0, \theta)$, including the Markov chain description due to Vershik, Shmidt and Ignatov, are generalized to the two-parameter case. The size-biased random permutation of $\mathsf{PD}(\alpha, \theta)$ 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 $0 < \alpha < 1, \mathsf{PD}(\alpha, 0)$ 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 $\alpha$. Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950s and 1960s. The distribution of ranked lengths of excursions of a one-dimensional Brownian motion is $\mathsf{PD}(1/2, 0)$, and the corresponding distribution for a Brownian bredge is $\mathsf{PD}(1/2, 1/2)$. The $\mathsf{PD}(\alpha, 0)$ and $\mathsf{PD}(\alpha, \alpha)$ distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable Markov process whose zero set is the range of a stable subordinator of index $\alpha$.
1,281 citations
TL;DR: In this article, explicit formulae for the distribution of various random partitions of a positive integer n, both ordered and unordered, derived from the zero set M of a Brownian motion by the following scheme: pick n points uniformly at random from [0, 1], and classify them by whether they fall in the same or different component intervals of the complement of M.
Abstract: Explicit formulae are obtained for the distribution of various random partitions of a positive integer n, both ordered and unordered, derived from the zero set M of a Brownian motion by the following scheme: pick n points uniformly at random from [0, 1], and classify them by whether they fall in the same or different component intervals of the complement of M. Corresponding results are obtained for M the range of a stable subordinator and for bridges defined by conditioning on 1 E M. These formulae are related to discrete renewal theory by a general method of discretizing a subordinator using the points of an independent homogeneous Poisson process.
102 citations
01 Jan 1997
TL;DR: In this paper, 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 is analyzed.
Abstract: 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 Levy measure of the Poisson point process of interval lengths on the local time scale. The martingales derived from the zero set of a one-dimensional diffusion are related to martingales studied by Azema and Rainer. Formulae are obtained which show how the distribution of interval lengths is affected when the underlying process is subjected to a Girsanov transoformation. In particular, results for the zero set of an Ornstein-Uhlenbeck process or a Cox-Ingersoll-Ross process are derived from results for a Brownian motion or recurrent Bessel process, when the zero set is the range of a stable subordinator.
29 citations
01 Jan 1997
TL;DR: For a large class of random times T the distribution of relative lengths prior to T is the same as if T were a fixed time as discussed by the authors, and absolute continuity relations are obtained which relate the law of the relative lengths at time T to the law at fixed time.
Abstract: 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 arc-sine 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.
29 citations
TL;DR: In this paper, a class of probability measures in the space of virtual permutations associated with subordinators (i.e., processes with stationary positive independent increments) are introduced, and they are shown to be quasi-invariant under both left and right actions of the countable symmetric group.
Abstract: We introduce a class of probability measures in the space of virtual permutations associated with subordinators (i.e., processes with stationary positive independent increments). We prove that these measures are quasiinvariant under both left and right actions of the countable symmetric group\(\mathfrak{S}^\infty \), and a simple formula for the corresponding cocycle is obtained. In case of a stable subordinator, we find the value of the spherical function of a constant vector on the class of transpositions. Bibliography: 19 titles.
12 citations
Book•
16 May 1997
TL;DR: In this paper, a differentiable isomorphism between Wiener space and path group is defined for the Riemannian manifold, and the convergence of Markov processes is studied.
Abstract: Branching processes, the Ray-Knight theorem, and sticky Brownian motion.- Integration by parts and Cameron-Martin formulas for the free path space of a compact Riemannian manifold.- The change of variables formula on Wiener space.- Classification des Semi-Groupes de diffusion sur IR associes a une famille de polynomes orthogonaux.- A differentiable isomorphism between Wiener space and path group.- On martingales which are finite sums of independent random variables with time dependent coefficients.- Oscillation presque sure de martingales continues.- A note on Cramer's theorem.- The hypercontractivity of Ornstein-Uhlenbeck semigroups with drift, revisited.- Une preuve standard du principe d'invariance de stoll.- Marches aleatoires auto-evitantes et mesures de polymere.- On the tails of the supremum and the quadratic variation of strictly local martingales.- On Wald's equation. Discrete time case.- Remarques sur l'hypercontractivite et l'evolution de l'entropie pour des chaines de Markov finies.- Comportement des temps d'atteinte d'une diffusion fortement rentrante.- Closed sets supporting a continuous divergent martingale.- Some polar sets for the Brownian sheet.- A counter-example concerning a condition of Ogawa integrability.- The multiplicity of stochastic processes.- Theoremes limites pour les temps locaux d'un processus stable symetrique.- An Ito type isometry for loops in Rd via the Brownian bridge.- On continuous conditional Gaussian martingales and stable convergence in law.- Simple examples of non-generating Girsanov processes.- Formule d'Ito generalisee pour le mouvement brownien lineaire.- On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman's theorem.- Some remarks on Pitman's theorem.- On the lengths of excursions of some Markov processes.- On the relative lengths of excursions derived from a stable subordinator.- Some remarks about the joint law of Brownian motion and its supremum.- A characterization of Markov solutions for stochastic differential equations with jumps.- Diffeomorphisms of the circle and the based stochastic loop space.- Vitesse de convergence en loi pour des solutions d'equations differentielles stochastiques vers une diffusion.- Projection d'une diffusion reelle sur sa filtration lente.
12 citations
TL;DR: In this article, a general subordinator, X, whose range up to time 1,X([0, 1]), has similar structure as random reorderings of the Cantor setK(ω), and has the same exact Hausdorff measure function and the integal test of packing measure.
Abstract: In this paper we have found a general subordinator,X, whose range up to time 1,X([0, 1)), has similar structure as random re-orderings of the Cantor setK(ω).X([0, 1)) andK(ω) have the same exact Hausdorff measure function and the integal test of packing measure.