Extremes of a certain class of Gaussian processes

• Published in: Stochastic processes and their applications Vol. 83; no. 2; pp. 257 - 271
• Main Authors: Hüsler, J., Piterbarg, V.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.10.1999; Elsevier
• Series: Stochastic Processes and their Applications
• Subjects: Extreme values; Extreme values Gaussian processes Fractional Brownian motions Self-similar processes; Fractional Brownian motions; Gaussian processes; Self-similar processes; Fractional Brownian motions; Self-similar processes; Gaussian processes; Extreme values
• ISSN: 0304-4149; 1879-209X
• DOI: 10.1016/S0304-4149(99)00041-1

We consider the extreme values of fractional Brownian motions, self-similar Gaussian processes and more general Gaussian processes which have a trend − ct β for some constants c, β>0 and a variance t 2 H . We derive the tail behaviour of these extremes and show that they occur mainly in the neighbourhood of the unique point t 0 where the related boundary function ( u+ ct β )/ t H is minimal. We consider the case that H< β.