Comptes Rendus
Probability Theory
Asymptotics for the distribution of lengths of excursions of a d-dimensional Bessel process (0<d<2)
[Asymptotiques pour la distribution des longueurs des excursions d'un processus de Bessel de dimension d (0<d<2)]
Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 201-208.

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.

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.

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DOI : 10.1016/j.crma.2006.06.010

Bernard Roynette 1 ; Pierre Vallois 1 ; Marc Yor 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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     author = {Bernard Roynette and Pierre Vallois and Marc Yor},
     title = {Asymptotics for the distribution of lengths of excursions of a \protect\emph{d}-dimensional {Bessel} process $ (0<d<2)$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {201--208},
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     year = {2006},
     doi = {10.1016/j.crma.2006.06.010},
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Bernard Roynette; Pierre Vallois; Marc Yor. Asymptotics for the distribution of lengths of excursions of a d-dimensional Bessel process $ (0
                  
                

[1] C. Donati-Martin, B. Roynette, P. Vallois, M. Yor, On constants related to the choice of the local time at 0 and the corresponding Itô measure for Bessel processes with dimension d=2(1α), Studia Math. Hung. (2006), in press

[2] B. De Meyer; B. Roynette; P. Vallois; M. Yor On independent times and positions for Brownian motions, Rev. Math. Iberoamericana, Volume 18 (2002) no. 3, pp. 541-586

[3] F.B. Knight On the duration of the longest excursion, Sem. Stoch. Prob., 1985, Birkhäuser, Basel, 1986, pp. 117-147

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[6] J. Pitman; M. Yor The two parameter Poisson–Dirichlet distribution derived from a stable subordinator, Ann. Probab., Volume 25 (1997) no. 2, pp. 855-900

[7] J. Pitman; M. Yor On the relative lengths of excursions derived from a stable subordinator, Sém. Probab. XXXI, Lecture Notes in Math., vol. 1655, 1997, pp. 287-305

[8] B. Roynette, P. Vallois, M. Yor, Penalizing a Brownian motion with a function of the lengths of its excursions, VII (March 2006), in preparation

[9] B. Roynette, P. Vallois, M. Yor, Penalisation of a Bessel process of dimension d=2(1α) (0<d<2) by a function of its longest excursion, IX (March 2006), in preparation

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