ESAIM: Probability and Statistics (ESAIM: P&S)

ESAIM: P&S, August 2007, Vol. 11, pp. 448-465
DOI: 10.1051/ps:2007029

The fractional mixed fractional brownian motion and fractional brownian sheet

Charles El-Nouty

U.F.R. de mathématiques, Université Paris VI, 175 rue du Chevaleret, 75013 Paris, France; elnouty@ccr.jussieu.fr

(Received September 29, 2005. Revised May 2, 2006. Published online 17 August 2007.)

Abstract

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.

Mathematics Subject Classification. 60F15, 60G15

Key words: Fractional Brownian motion, fractional Brownian sheet, lower classes, small ball probabilities.

© EDP Sciences, SMAI 2007