Comptes Rendus
no. 6
Probability Theory/Mathematical Analysis
Local and asymptotic properties of Linear Fractional Stable Sheets
[Comportement local et asymptotique du drap linéaire fractionnaire stable]

Présenté par : Yves Meyer

Antoine Ayache1 ; François Roueff2 ; Yimin Xiao3
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. 6, pp. 389-394.

Dans cette Note, nous introduisons une représentation du drap linéaire fractionnaire stable sous la forme d'une série aléatoire d'ondelettes. Au moyen de cette représentation, dans le cas où les trajectoires du processus sont continues, un module de continuité anisotropique uniforme quasi-optimal de ces trajectoires est obtenu ainsi qu'un contrôle de leur comportement à l'infini.

In this Note we introduce representations of Linear Fractional Stable Sheets as wavelet random series. Using these representations, in the case where the paths are continuous, an anisotropic uniform and quasi-optimal modulus of continuity of these paths is obtained as well as an upper bound on their behavior at infinity.

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Accepté le :
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DOI : 10.1016/j.crma.2007.01.017
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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     title = {Local and asymptotic properties of {Linear} {Fractional} {Stable} {Sheets}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {389--394},
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Antoine Ayache; François Roueff; Yimin Xiao. Local and asymptotic properties of Linear Fractional Stable Sheets. Comptes Rendus. Mathématique, Volume 344 (2007) no. 6, pp. 389-394. doi : 10.1016/j.crma.2007.01.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.01.017/

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

[2] A. Benassi; D. Roux Elliptic self-similar stochastic processes, Rev. Mat. Iberoamericana, Volume 19 (2003), pp. 767-796

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

[4] I. Daubechies Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992

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

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

[7] Y. Meyer Wavelets and Operators, Cambridge University Press, Cambridge, UK, 1992

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

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

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