Asymptotics of Maxima of Strongly Dependent Gaussian Processes

Abstract

Let {Xn(t),t∈[0,∞)},n∈ℕ, be standard stationary Gaussian processes. The limit distribution oft∈[0,T(n)]|Xn(t)| is established asrn(t), the correlation function of {Xn(t),t∈[0,∞)},n∈ℕ, which satisfies the local and long-range strong dependence conditions, extending the results obtained in Seleznjev (1991).

Piterbarg's max-discretization theorem for stationary vector Gaussian processes observed on different grids

“…b) The Berman condition is relaxed by assuming that (3) holds for some r ∈ [0, ∞). When r > 0 the Gaussian process X is said to be strongly dependent, see [22,26,23,32,29,8] for details on the extremes of such Gaussian processes. The contribution [34] derives Piterbarg's max-discretisation theorem for strongly dependent Gaussian processes.…”

Exact tail asymptotics of the supremum of strongly dependent Gaussian processes over a random interval

“…Indeed, condition (A3) is a natural extension of (A2). For related studies on extremes of strongly dependent Gaussian processes, we refer to [24], [28], [16], [18], [19], [30], [15], [31], [32], [33], [34]. The aim of this paper is to study the asymptotic behaviour of the supremum of strongly dependent stationary Gaussian processes over some random interval [0, T ].…”

On Piterbarg's max-discretisation theorem for homogeneous Gaussian random fields

“…Pickands (1969), Leadbetter et al (1983), Piterbarg (1996). The extensions of the classic result (3) to more general cases, such as for non-stationary case, strongly dependent case, can be found in Mittal and Ylvisaker (1975), McCormick (1980), McCormick and Qi (2000), Hülser (1990), Konstant and Piterbarg (1993), Seleznjev (1991Seleznjev ( , 1996, Hülser (1999), Hülser et al (2003), Tan et al (2012) and among others. In applied fields, however, the classic result (3) can not be used directly, since the available samples are discrete.…”

“…[15,11,16]. The extensions of the classic result (3) to more general cases, such as for non-stationary case, strongly dependent case, can be found in [14,12,13,5,10,18,19,6,9,23] and among others.…”