Probability Theory
Asymptotics for the distribution of lengths of excursions of a d-dimensional Bessel process (0<d<2)
Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 201-208.

Let (Rt,t0) denote a d-dimensional Bessel process (0<d<2). For every t0, we consider the times gt=sup{st:Rs=0}, and dt=inf{s>t:Rs=0}, as well as the three sequences: (Vgtn,n1), (Vtn,n2), and (Vdtn,n2), which consist of the lengths of excursions of R away from 0 before gt, before t, and before dt, respectively, each one being ranked by decreasing order.

We obtain a limit theorem concerning each of the laws of these three sequences, as t. The result is expressed in terms of a positive, σ-finite measure Π on the set S of decreasing sequences. Π is closely related with the Poisson–Dirichlet laws on S.

Soit (Rt,t0) un processus de Bessel de dimension d(0,2). Pour tout t0, on considère les temps gt=sup{st:Rs=0} et dt=inf{s>t:Rs=0}, ainsi que les trois suites : (Vgtn,n1), resp. (Vtn,n2), resp. (Vdtn,n2) des longueurs d'excursions de R hors de 0, avant gt, resp. avant t, resp. avant dt, rangées par ordre décroissant.

Nous obtenons un théorème limite concernant chacune des lois de ces trois suites, lorsque t. Ce théorème s'exprime à l'aide d'une mesure positive, σ-finie, Π sur S={s=(s1,s2,,sn,);s1s2sn0}. Π est intimement liée aux lois de Poisson–Dirichlet sur S.

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DOI: 10.1016/j.crma.2006.06.010
Roynette, Bernard 1; Vallois, Pierre 1; Yor, Marc 2, 3

1 Université Henri-Poincaré, Institut Elie-Cartan, BP239, 54506 Vandoeuvre-les-Nancy cedex, France
2 Laboratoire de probabilités et modèles aléatoires, Universités Paris VI et VII, 4, place Jussieu, case 188, 75252 Paris cedex 05, France
3 Institut Universitaire de France, France
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Roynette, Bernard; Vallois, Pierre; Yor, Marc. Asymptotics for the distribution of lengths of excursions of a d-dimensional Bessel process $ (0
                  
                

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