Discrete and Continuous Time Extremes of Gaussian Processes

Abstract

Joint distribution of maxima of a Gaussian stationary process on a continuous time and in uniform grid on the real axis is studied. When the grid is sufficiently sparse, maxima are asymptotically independent. When the grid is sufficiently tight, the maxima asymptotically coincide. In the boundary case which we call Pickands' grid, the limit distribution is non-degenerate. It calculated in terms of a Pickands' type constant.

On Piterbarg Max-Discretisation Theorem for Standardised Maximum of Stationary Gaussian Processes

“…Piterbarg (2004) first studied the asymptotic relation between M T and the maximum of the discrete version Hüsler (2004) and Piterbarg (2004), a uniform grid R = R(δ) = {kδ :…”

“…Under the above setting, the deep paper Piterbarg (2004) showed that the maximum M δ T taken over discrete time points and the maximum M T of the continuous time points can be asymptotically independent, dependent or totally dependent if the grid is a sparse, a Pickands or a dense grid, respectively. We refer to that result of Piterbarg (2004) as Piterbarg max-discretisation theorem.…”

“…

Abstract: With motivation from Hüsler (2004) and Piterbarg (2004) in this paper we derive the joint limiting distribution of standardised maximum of a continuous, stationary Gaussian process and the standardised maximum of this process sampled at discrete time points. We prove that these two random sequences are asymptotically complete dependent if the grid of the discrete time points is sufficiently dense, and asymptotically independent if the grid is sufficiently sparse.

…”

Generalized Pickands constants and stationary max-stable processes

“…Recently, the seminal contribution Dieker and Yakir (2014) The principal advantage of Dieker-Yakir representation (3) is that it is given as an expectation rather than as a limit, which is particularly useful for Monte Carlo simulations of H δ W . Pickands constants traditionally also appear in Gumbel limit theorems, see e.g., Berman (1992); Piterbarg (2004). Such limit theorems are recently formulated for max-stable processes and provide a first link of classical Gaussian tail asymptotics to spatial extreme value theory.…”

Dependence Between Extreme Values of Discrete and Continuous Time Locally Stationary Gaussian Processes

“…For a stationary Gaussian process, with "(t) = 0 and '(t) = 1, Piterbarg (2004) shows that the maximum M ð Þ T of discrete time points and the maximum M T of the continuous time points can be asymptotically independent, dependent or totally dependent if the grid if a sparse, a Pickands or a dense grid, respectively. The definitions of these grids are as follows, related to the normalization for the maximum M T .…”