# On Generalized Berman Constants

Chengxiu Ling () and Hong Zhang ()
Chengxiu Ling: Xi’an Jiaotong-Liverpool University
Hong Zhang: Southwest University

Methodology and Computing in Applied Probability, 2020, vol. 22, issue 3, 1125-1143

Abstract: 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 B α h ( x , E ) = ∫ ℝ e z ℙ ∫ E I 2 B α ( t ) − | t | α − h ( t ) − z > 0 d t > x d z , x ∈ [ 0 , mes ( E ) ] , $$\mathcal{B}_{\alpha}^{h}(x, E) = {\int}_{\mathbb{R}} e^{z} \mathbb{P} \left\{{{\int}_{E} \mathbb{I}\left( \sqrt2B_{\alpha}(t) - |t|^{\alpha} - h(t) - z>0 \right) \text{d} t \!>\! x}\right\} \text{d} z,\quad x\in[0, \text{mes}(E)],$$ where mes(E) is the Lebesgue measure of a compact set E ⊂ ℝ $E\subset \mathbb {R}$ , h is a continuous drift function, and Bα 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 B 2 h ( x , E ) ${{\mathcal{B}}_{2}^{h}}(x, E)$ with typical drift functions are given and several bounds of B α h ( x , E ) ${\mathcal{B}}_{\alpha }^{h}(x, E)$ are established as well. Numerical studies are performed to illustrate the main results.

Keywords: Berman constants; Sojourn time; Fractional Brownian motion; Gaussian process; Primary 60G15; Secondary 60G70 (search for similar items in EconPapers)
Date: 2020
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