On Generalized Berman Constants

Abstract

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 by

$$ \mathcal{B}_{\alpha}^{h}(x, E) = {\int}_{\mathbb{R}} e^{z} \mathbb{P} \left\{{{\int}_{E} \mathbb{I}\left( \sqrt2B_{\alpha}(t) - |t|^{\alpha} - h(t) - z>0 \right) \text{d} t \!>\! x}\right\} \text{d} z,\quad x\in[0, \text{mes}(E)], $$

where mes(E) is the Lebesgue measure of a compact set $E\subset \mathbb {R}$, 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 ${{\mathcal{B}}_{2}^{h}}(x, E)$ with typical drift functions are given and several bounds of ${\mathcal{B}}_{\alpha }^{h}(x, E)$ are established as well. Numerical studies are performed to illustrate the main results.