A note on upper estimates for Pickands constants

Pickands constants  HBαHBα play a significant role in the extreme value theory of Gaussian processes. Recall that HBα≔limT→∞Eexp(supt∈[0,T](2Bα(t)−tα))T, where {Bα(t),t≥0}{Bα(t),t≥0} is a fractional Brownian motion with Hurst parameter α/2α/2 and α∈(0,2]α∈(0,2].In this note we derive new upper bounds for HBαHBα and α∈(1,2]α∈(1,2]. The obtained results improve bounds given by Shao [Shao, Q.M., 1996. Bounds and estimators of a basic constant in extreme value theory of Gaussian processes. Statist. Sinica 6, 245–257].