Asymptotics for the distribution of lengths of excursions of a <i>d</i>-dimensional Bessel process $ (0<d<2)$

Bernard Roynette; Pierre Vallois; Marc Yor

doi:10.1016/j.crma.2006.06.010

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

)]

Présenté par : Marc Yor

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

Comptes Rendus. Mathématique, Volume 343 (2006) no. 3, pp. 201-208.

Résumé
Abstract

Soit $(R_{t}, t ⩾ 0)$ un processus de Bessel de dimension $d \in (0, 2)$ . Pour tout $t ⩾ 0$ , on considère les temps $g_{t} = \sup {s ⩽ t : R_{s} = 0}$ et $d_{t} = \inf {s > t : R_{s} = 0}$ , ainsi que les trois suites : $(V_{g_{t}}^{n}, n ⩾ 1)$ , resp. $(V_{t}^{n}, n ⩾ 2)$ , resp. $(V_{d_{t}}^{n}, n ⩾ 2)$ des longueurs d'excursions de R hors de 0, avant $g_{t}$ , resp. avant t, resp. avant $d_{t}$ , rangées par ordre décroissant.

Nous obtenons un théorème limite concernant chacune des lois de ces trois suites, lorsque $t \to \infty$ . Ce théorème s'exprime à l'aide d'une mesure positive, σ-finie, Π sur $S^{↓} = {s = (s_{1}, s_{2}, \dots, s_{n}, \dots); s_{1} ⩾ s_{2} ⩾ \dots ⩾ s_{n} ⩾ \dots ⩾ 0}$ . Π est intimement liée aux lois de Poisson–Dirichlet sur $S^{↓}$ .

Let $(R_{t}, t ⩾ 0)$ denote a d-dimensional Bessel process $(0 < d < 2)$ . For every $t ⩾ 0$ , we consider the times $g_{t} = \sup {s ⩽ t : R_{s} = 0}$ , and $d_{t} = \inf {s > t : R_{s} = 0}$ , as well as the three sequences: $(V_{g_{t}}^{n}, n ⩾ 1)$ , $(V_{t}^{n}, n ⩾ 2)$ , and $(V_{d_{t}}^{n}, n ⩾ 2)$ , which consist of the lengths of excursions of R away from 0 before $g_{t}$ , before t, and before $d_{t}$ , respectively, each one being ranked by decreasing order.

We obtain a limit theorem concerning each of the laws of these three sequences, as $t \to \infty$ . 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 $S^{↓}$ .

Reçu le : 2006-03-24
Accepté le : 2006-05-29
Publié le : 2006-07-11

DOI : 10.1016/j.crma.2006.06.010

Affiliations des auteurs :

Bernard Roynette ¹ ; Pierre Vallois ¹ ; Marc Yor ^{2, 3}

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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},
     publisher = {Elsevier},
     volume = {343},
     number = {3},
     year = {2006},
     doi = {10.1016/j.crma.2006.06.010},
     language = {en},
}

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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

Bibliographie
Cité par

[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 $d = 2 (1 - α)$ , 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

[4] N.N. Lebedev Special Functions and their Applications, Dover Pub. Inc., New York, 1965

[5] J. Pitman; M. Yor Arc sine laws and interval partitions derived from a stable subordinator, Proc. London Math. Soc., Volume 65 (1992) no. 3, pp. 326-356

[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 $d = 2 (1 - α)$ $(0 < d < 2)$ by a function of its longest excursion, IX (March 2006), in preparation

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