A note on upper estimates for Pickands constants

Abstract

Pickands constants H B α play a significant role in the extreme value theory of Gaussian processes. Recall that H B α ≔ lim T → ∞ E exp ( sup t ∈ [ 0 , T ] ( 2 B α ( t ) − t α ) ) T , where { B α ( t ) , t ≥ 0 } is a fractional Brownian motion with Hurst parameter α / 2 and α ∈ ( 0 , 2 ] . In this note we derive new upper bounds for H B α and 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].