The fractional mixed fractional brownian motion and fractional brownian sheet
ESAIM: Probability and Statistics, Tome 11 (2007), pp. 448-465.

We introduce the fractional mixed fractional brownian motion and fractional brownian sheet, and investigate the small ball behavior of its sup-norm statistic. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.

DOI : 10.1051/ps:2007029
Classification : 60F15, 60G15
Mots clés : fractional brownian motion, fractional brownian sheet, lower classes, small ball probabilities 
@article{PS_2007__11__448_0,
     author = {El-Nouty, Charles},
     title = {The fractional mixed fractional brownian motion and fractional brownian sheet},
     journal = {ESAIM: Probability and Statistics},
     pages = {448--465},
     publisher = {EDP-Sciences},
     volume = {11},
     year = {2007},
     doi = {10.1051/ps:2007029},
     mrnumber = {2339303},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps:2007029/}
}
TY  - JOUR
AU  - El-Nouty, Charles
TI  - The fractional mixed fractional brownian motion and fractional brownian sheet
JO  - ESAIM: Probability and Statistics
PY  - 2007
SP  - 448
EP  - 465
VL  - 11
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ps:2007029/
DO  - 10.1051/ps:2007029
LA  - en
ID  - PS_2007__11__448_0
ER  - 
%0 Journal Article
%A El-Nouty, Charles
%T The fractional mixed fractional brownian motion and fractional brownian sheet
%J ESAIM: Probability and Statistics
%D 2007
%P 448-465
%V 11
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/ps:2007029/
%R 10.1051/ps:2007029
%G en
%F PS_2007__11__448_0
El-Nouty, Charles. The fractional mixed fractional brownian motion and fractional brownian sheet. ESAIM: Probability and Statistics, Tome 11 (2007), pp. 448-465. doi : 10.1051/ps:2007029. http://www.numdam.org/articles/10.1051/ps:2007029/

[1] A. Ayache, S. Leger and M. Pontier, Drap Brownien fractionnaire. Potential Anal. 178 (2002) 31-43. | MR | Zbl

[2] A. Ayache and Y. Xiao, Asymptotic properties and Hausdorff dimensions of fractional Brownian sheets. J. Fourier Anal. Appl. 11 (2005) 407-439. | MR | Zbl

[3] E. Belinsky and W. Linde, Small Ball Probabilities of Fractional Brownian Sheets via Fractional Integration Operators. J. Theoret. Probab. 15 (2002) 589-612. | MR | Zbl

[4] C. Borell, Convex measures on locally convex space. Math. Ark. Math. 12 (1974) 239-252. | Zbl

[5] P. Cheridito, Mixed fractional Brownian motion. Bernoulli 7 (2001) 913-934. | Zbl

[6] N.J. Cutland, P.E. Kopp and W. Willinger, Stock price returns and the Joseph effect: a fractional version of the Black-Scholes model. in Seminar on Stochastic Analysis, Random Fields and Applications, Progr. Probab. E. Bolthausen, M. Dozziand F. Russo Eds., Basel: Birkhauser 36 (1995) 327-351. | Zbl

[7] C. El-Nouty, On the lower classes of fractional Brownian motion. Studia Sci. Math. Hungar. 37 (2001) 363-390. | Zbl

[8] C. El-Nouty, Lower classes of fractional Brownian motion under Hölder norms, Limit Theorems in Probability and Statistics, Balatonlelle, 1999, I. Berkes, E. Csáki, M. Csörgő Eds., János Bolyai Mathematical Society, Budapest (2002) 7-34. | Zbl

[9] C. El-Nouty, The fractional mixed fractional Brownian motion. Statist. Probab. Lett. 65 (2003) 111-120. | Zbl

[10] C. El-Nouty, Lower classes of integrated fractional Brownian motion. Studia Sci. Math. Hungar. 41 (2004) 17-38. | Zbl

[11] C. El-Nouty, The influence of a log-type small ball factor in the study of the lower classes. Bull. Sci. math. 129 (2005) 318-338. | Zbl

[12] F. Gassmann and D. Bürki, Experimental investigation of atmospheric dispersion over the Swiss Plain - Experiment SIESTA, Boundary-Layer Meteorology, Springer Netherlands 41 (1987) 295-307.

[13] F. Gassmann, P. Gaglione, S.E. Gryning, H. Hasenjäger, E. Lyck, H. Richner, B. Neiniger, S. Vogt and P. Thomas, Experimental Investigation of Atmospheric Dispersion over Complex Terrain in a Prealpine Region (experiment SIESTA) Swiss Federal Institute for Reactor Research EIR 604 (1986).

[14] T. Kühn and W. Linde, Optimal series representation of fractional Brownian sheets. Bernoulli 8 (2002) 669-696. | Zbl

[15] M. Ledoux and M. Talagrand, Probability in Banach spaces. Springer Verlag, Berlin (1994). | MR | Zbl

[16] W.V. Li, A Gaussian correlation inequality and its applications to small ball probabilities. Elect. Comm. in Probab. 4 (1999) 111-118. | Zbl

[17] W.V. Li and W. Linde, Existence of small ball constants for fractional Brownian motions. C. R. Acad. Sci. Paris 326 (1998) 1329-1334. | Zbl

[18] W.V. Li and Q.M. Shao, Gaussian Processes: Inequalities, Small Ball Probabilities and Applications, Stochastic Processes: Theory and Methods, Handbook of Statistics 19 (2001). | MR | Zbl

[19] M.A. Lifshits, Gaussian Random Functions. Kluwer Academic Publishers, Dordrecht (1995). | MR | Zbl

[20] D.M. Mason and Z. Shi, Small Deviations for Some Multi-Parameter Gaussian Processes. J. Theoret. Probab. 14 (2001) 213-239. | Zbl

[21] D. Monrad and H. Rootzen, Small values of Gaussian processes and functional laws of the iterated logarithm. Probab. Theory Related Fields 101 (1995) 173-192. | Zbl

[22] P. Révész, Random walk in random and non-random environments, World Scientific Publishing Co., Teaneck, NJ (1990). | MR | Zbl

[23] P.A. Samuelson, Rational theory of warrant pricing. Indust. Management Rev. 6 (1965) 13-31.

[24] M. Talagrand, Lower classes of fractional Brownian motion. J. Theoret. Probab. 9 (1996) 191-213. | Zbl

[25] Y. Xiao and T. Zhang, Local times of fractional Brownian sheets. Probab. Theory Related Fields 124 (2002) 204-226. | Zbl

Cité par Sources :