Joint continuity of the local times of linear fractional stable sheets

Antoine Ayache; François Roueff; Yimin Xiao

doi:10.1016/j.crma.2007.03.028

Probability Theory

Joint continuity of the local times of linear fractional stable sheets

Presented by: Jean-Pierre Kahane

Antoine Ayache ¹; François Roueff ²; Yimin Xiao ³

¹ UMR CNRS 8524, laboratoire Paul-Painlevé, bâtiment M2, Université Lille 1, 59655 Villeneuve d'Ascq cedex, France
² Télécom Paris/CNRS LTCI, 46, rue Barrault, 75634 Paris cedex 13, France
³ Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA

Comptes Rendus. Mathématique, Volume 344 (2007) no. 10, pp. 635-640.

Abstract
Résumé

Linear fractional stable sheets (LFSS) are a class of random fields containing the class of fractional Brownian sheets (FBS) by allowing, in the linear fractional representation of the FBS, the random measure to be α-stable with $α \in (0, 2]$ . In this Note, we extend some properties of the local time shown in the Gaussian case to the symmetric α-stable case. For any $N ⩾ 1$ , an $(N, 1)$ -LFSS is a real valued random field defined on $R_{+}^{N}$ . When $N = 1$ , the process is called linear fractional stable motion (LFSM). For $N ⩾ 1$ , an $(N, 1)$ -LFSS is mainly parameterized by a multidimensional index $H = (H_{1}, \dots, H_{N}) \in {(0, 1)}^{N}$ . Let $N, d ⩾ 1$ be fixed, we consider a random field defined on $R_{+}^{N}$ and taking its values in $R^{d}$ , an $(N, d)$ -LFSS, whose components are d independent copies of the same $(N, 1)$ -LFSS. We show that, if $d < H_{1}^{−1} + \dots + H_{N}^{−1}$ , then the $(N, d)$ -LFSS with index H has a local time. Moreover, when the sample path of the LFSS is continuous, that is, for $α < 2$ , when $H_{1}, \dots, H_{N} > 1 / α$ , we show that the local time is jointly continuous.

Le drap linéaire fractionnaire stable (LFSS) est un champ aléatoire qui généralise le drap brownien fractionnaire en remplaçant la mesure gaussienne dans sa représentation linéaire fractionnaire par une mesure α-stable, $α \in (0, 2]$ . Dans cette Note nous étendons certaines propriétés du temps local montrées pour le cas gaussien au cas symétrique α-stable. Le $(N, 1)$ -LFSS, $N ⩾ 1$ , est défini sur $R_{+}^{N}$ et prend ses valeurs dans $R$ , le cas $N = 1$ correspondant au mouvement linéaire fractionnaire stable (LFSM). Ce champ est principalement paramétré par $H = (H_{1}, \dots, H_{N}) \in {(0, 1)}^{N}$ . Nous considérons un champ aléatoire à valeurs dans $R^{d}$ , le $(N, d)$ -LFSS, $N, d ⩾ 1$ , défini en prenant d copies indépendantes d'un $(N, 1)$ -LFSS. Nous montrons que, si $d < H_{1}^{−1} + \dots + H_{N}^{−1}$ , alors le $(N, d)$ -LFSS de paramètre H admet un temps local. De plus, dans le cas où ses trajectoires sont continues, i.e., pour $α < 2$ , quand les paramètres vérifient $H_{1}, \dots, H_{N} > 1 / α$ , nous établissons la bicontinuité de ce temps local.

Received: 2006-10-02
Accepted: 2007-03-23
Published online: 2007-05-10

DOI: 10.1016/j.crma.2007.03.028

Author's affiliations:

Antoine Ayache ¹; François Roueff ²; Yimin Xiao ³

¹ UMR CNRS 8524, laboratoire Paul-Painlevé, bâtiment M2, Université Lille 1, 59655 Villeneuve d'Ascq cedex, France
² Télécom Paris/CNRS LTCI, 46, rue Barrault, 75634 Paris cedex 13, France
³ Department of Statistics and Probability, Michigan State University, East Lansing, MI 48824, USA

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     author = {Antoine Ayache and Fran\c{c}ois Roueff and Yimin Xiao},
     title = {Joint continuity of the local times of linear fractional stable sheets},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {635--640},
     publisher = {Elsevier},
     volume = {344},
     number = {10},
     year = {2007},
     doi = {10.1016/j.crma.2007.03.028},
     language = {en},
}

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Antoine Ayache; François Roueff; Yimin Xiao. Joint continuity of the local times of linear fractional stable sheets. Comptes Rendus. Mathématique, Volume 344 (2007) no. 10, pp. 635-640. doi : 10.1016/j.crma.2007.03.028. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.03.028/

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