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

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Abstract

Let {BH1,H2(t1, t2), t1≥ 0, t2≥ 0} be a fractional Brownian sheet with indexes 0 < H1, H2< 1. When H1= H2:= H, there is a logarithmic factor in the small ball function of the sup-norm statistic of BH,H. First, we state general conditions (one based on a logarithmic factor in the small ball function) on some statistics of BH,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. © 2004 Elsevier SAS. All rights reserved.

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El-Nouty, C. (2005). The influence of a log-type small ball factor in the study of the lower classes. Bulletin Des Sciences Mathematiques, 129(4), 318–338. https://doi.org/10.1016/j.bulsci.2004.11.004

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