The influence of a log-type small ball factor in the study of the lower classes

Abstract

Let { B H 1 , H 2 ( t 1 , t 2 ) , t 1 ⩾ 0 , t 2 ⩾ 0 } be a fractional Brownian sheet with indexes 0 < H 1 , H 2 < 1 . When H 1 = H 2 : = H , there is a logarithmic factor in the small ball function of the sup-norm statistic of B H , H . First, we state general conditions (one based on a logarithmic factor in the small ball function) on some statistics of B H , H . Then we characterize the sufficiency part of the lower classes of these statistics by an integral test. Finally, when we consider the sup-norm statistic, the influence of the log-type small ball factor in the necessity part is measured by a second integral test.