Journal of Applied Mathematics and Stochastic Analysis 
Volume 2008 (2008), Article ID 160303, 16 pages
doi:10.1155/2008/160303
Research Article

On the Lower Classes of Some Mixed Fractional Gaussian Processes with Two Logarithmic Factors

Charles El-Nouty

 U.F.R. de Mathématiques, Université Pierre et Marie Curie, 175 rue du Chevaleret, Paris 75013, France
Received 5 February 2007; Revised 10 July 2007; Accepted 12 September 2007

Recommended by Hideo Nagai

Abstract

We introduce the fractional mixed fractional Brownian sheet and investigate the small ball behavior of its sup-norm statistic by establishing a general result on the small ball probability of the sum of two not necessarily independent joint Gaussian random vectors. 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.