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

With motivation from Husler (Extremes 7:179-190, 2004) and Piterbarg (Extremes 7:161-177, 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. We show that our results are relevant for computational problems related to discrete time approximation of the continuous time maximum