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

Abstract

The maximum of a continuous, locally stationary Gaussian process which satisfies Berman's condition on the long range dependence is compared with the maximum of this process sampled at discrete time points. These two extreme values are asymptotically totally dependent if the grid of the discrete time points is sufficiently dense, and asymptotically independent if the the grid points are sparse.

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

“…This paper highlights the role of different grids in the approximation of the maximum over a continuous interval. Our results are therefore of interest for simulation studies, which was the main motivation of [27,19,20,36,37,30,35]. c) As a by-product we show that for weakly dependent stationary Gaussian processes the limiting distributions are max-stable.…”

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 :…”

“…Our calculations show that it is however much more difficult to obtain similar results to Theorem 1.1 for the studentised maximum. Additionally, at present it seems also very difficult to obtain results for Pickands grids, which has been the case also in Hüsler (2004). Due to those difficulties the aforementioned cases shall be treated in a forthcoming research paper.…”

“…

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.

…”

On maxima of chi-processes over threshold dependent grids

“…

In this paper, with motivation from [30] and the considerable interest in stationary chi-processes, we derive asymptotic joint distributions of maxima of stationary strongly dependent chi-processes on a continuous time and an uniform grid on the real axis. Our findings extend those for Gaussian cases and give three involved dependence structures via the strongly dependence condition and the sparse, Pickands and dense grids.The impetus for this investigation comes from numerical simulations of high extremes of continuous time random processes, see e.g., [15,30,37] for Gaussian processes, [16] for the storage process with fractional Brownian motion, [13,38,39] for stationary vector Gaussian processes and standardized stationary Gaussian processes, and [41] for stationary processes. It is shown in the aforementioned contributions that the dependence between continuous time extremes and discrete time extremes is determined strongly by the sampling frequency δ and the normalization constants, see also for related discussions [5,20,31,32,41] in the financial and time series literature.

…”