TY - JOUR
T1 - On Generalized Berman Constants
AU - Ling, Chengxiu
AU - Zhang, Hong
N1 - Publisher Copyright:
© 2019, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2020/9/1
Y1 - 2020/9/1
N2 - 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 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 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.
AB - 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 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 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.
KW - Berman constants
KW - Fractional Brownian motion
KW - Gaussian process
KW - Sojourn time
UR - http://www.scopus.com/inward/record.url?scp=85076604528&partnerID=8YFLogxK
U2 - 10.1007/s11009-019-09754-0
DO - 10.1007/s11009-019-09754-0
M3 - Article
AN - SCOPUS:85076604528
SN - 1387-5841
VL - 22
SP - 1125
EP - 1143
JO - Methodology and Computing in Applied Probability
JF - Methodology and Computing in Applied Probability
IS - 3
ER -