Probability Theory
Joint continuity of the local times of linear fractional stable sheets
Comptes Rendus. Mathématique, Volume 344 (2007) no. 10, pp. 635-640.

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(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 N1, 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 N1, an (N,1)-LFSS is mainly parameterized by a multidimensional index H=(H1,,HN)(0,1)N. Let N,d1 be fixed, we consider a random field defined on R+N and taking its values in Rd, an (N,d)-LFSS, whose components are d independent copies of the same (N,1)-LFSS. We show that, if d<H1−1++HN−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 H1,,HN>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(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, N1, 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=(H1,,HN)(0,1)N. Nous considérons un champ aléatoire à valeurs dans Rd, le (N,d)-LFSS, N,d1, défini en prenant d copies indépendantes d'un (N,1)-LFSS. Nous montrons que, si d<H1−1++HN−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 H1,,HN>1/α, nous établissons la bicontinuité de ce temps local.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2007.03.028
Ayache, Antoine 1; Roueff, François 2; Xiao, Yimin 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
@article{CRMATH_2007__344_10_635_0,
     author = {Ayache, Antoine and Roueff, Fran\c{c}ois and Xiao, Yimin},
     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},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2007.03.028/}
}
TY  - JOUR
AU  - Ayache, Antoine
AU  - Roueff, François
AU  - Xiao, Yimin
TI  - Joint continuity of the local times of linear fractional stable sheets
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 635
EP  - 640
VL  - 344
IS  - 10
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2007.03.028/
DO  - 10.1016/j.crma.2007.03.028
LA  - en
ID  - CRMATH_2007__344_10_635_0
ER  - 
%0 Journal Article
%A Ayache, Antoine
%A Roueff, François
%A Xiao, Yimin
%T Joint continuity of the local times of linear fractional stable sheets
%J Comptes Rendus. Mathématique
%D 2007
%P 635-640
%V 344
%N 10
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2007.03.028/
%R 10.1016/j.crma.2007.03.028
%G en
%F CRMATH_2007__344_10_635_0
Ayache, Antoine; Roueff, François; Xiao, Yimin. 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. http://www.numdam.org/articles/10.1016/j.crma.2007.03.028/

[1] Ayache, A.; Roueff, F.; Xiao, Y. Local and asymptotic properties of linear fractional stable sheets, C. R. Acad. Sci. Paris, Ser. I, Volume 344 (2007) no. 6, pp. 389-394

[2] A. Ayache, D. Wu, Y. Xiao, Joint continuity of the local times of fractional Brownian sheets, Ann. Inst. H. Poincaré Probab. Statist. (2006)

[3] Ayache, A.; Xiao, Y. Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets, J. Fourier Anal. Appl., Volume 11 (2005), pp. 407-439

[4] Bonami, A.; Estrade, A. Anisotropic analysis of some Gaussian models, J. Fourier Anal. Appl., Volume 9 (2003), pp. 215-236

[5] Ehm, W. Sample function properties of multi-parameter stable processes, Z. Wahrsch. Verw Gebiete, Volume 56 (1981), pp. 195-228

[6] Geman, D.; Horowitz, J. Occupation densities, Ann. Probab., Volume 8 (1980), pp. 1-67

[7] Kahane, J.-P. Some Random Series of Functions, Cambridge University Press, 1985

[8] Khoshnevisan, D. Multiparameter Processes: An Introduction to Random Fields, Springer, New York, 2002

[9] Kôno, N.; Maejima, M. Hölder continuity of sample paths of some self-similar stable processes, Tokyo J. Math., Volume 14 (1991), pp. 93-100

[10] Kôno, N.; Shieh, N.-R. Local times and related sample path properties of certain self-similar processes, J. Math. Kyoto Univ., Volume 33 (1993), pp. 51-64

[11] Maejima, M. A self-similar process with nowhere bounded sample paths, Z. Wahrsch. Verw. Gebiete, Volume 65 (1983), pp. 115-119

[12] Nolan, J. Local nondeterminism and local times for stable processes, Probab. Theory Related Fields, Volume 82 (1989), pp. 387-410

[13] Samorodnitsky, G.; Taqqu, M.S. Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman & Hall, New York, 1994

[14] Takashima, K. Sample path properties of ergodic self-similar processes, Osaka Math. J., Volume 26 (1989), pp. 159-189

[15] Walsh, J. The local time of the Brownian sheet, SMF Astérisque, Volume 52–53 (1978), pp. 47-61

[16] Xiao, Y.; Zhang, T. Local times of fractional Brownian sheets, Probab. Theory Related Fields, Volume 124 (2002), pp. 204-226

Cited by Sources: