On Generalized Berman Constants

Considering the important role in Gaussian related extreme value topics, we evaluate the Berman constants involved in the study of the sojourn time of Gaussian processes, given byBαh(x,E)=∫ℝezℙ{∫EI(2Bα(t)−|t|α−h(t)−z>0)dt>x}dz,x∈[0,mes(E)], where mes(E) is the Lebesgue measure of a compact set E⊂ ℝ, h is a continuous drift function, and B_α is a centered fractional Brownian motion (fBm) with Hurst index α/2 ∈ (0, 1]. This note specifies its explicit expression for α = 1 and α = 2 under certain conditions of drift functions. Explicit expressions of B2h(x,E) with typical drift functions are given and several bounds of Bαh(x,E) are established as well. Numerical studies are performed to illustrate the main results.