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

Abstract

In this paper we derive Piterbarg's max-discretisation theorem for two different grids considering centered stationary vector Gaussian processes. So far in the literature results in this direction have been derived for the joint distribution of the maximum of Gaussian processes over [0, T ] and over a grid R(δ 1 (T )) = {kδ 1 (T ) : k = 0, 1, · · · }. In this paper we extend the recent findings by considering additionally the maximum over another grid R(δ 2 (T )). We derive the joint limiting distribution of maximum of stationary Gaussian vector processes for different choices of such grids by letting T → ∞. As a by-product we find that the joint limiting distribution of the maximum over different grids, which we refer to as the Piterbarg distribution, is in the case of weakly dependent Gaussian processes a max-stable distribution.

On maxima of chi-processes over threshold dependent grids

“…(b) From our results we see that the joint convergence is determined by the choice of the grids and the normalization constants a T , b T and b δ,T , which is helpful in simulation studies and statistical applications, see related discussions for vector Gaussian processes in [13].…”

“…Results for extremes of chi-type processes for such generalizations can be found in [1,4,24,25]. (e) It might be interesting to investigate the limit theorems for different grids as in [13]. Another possibility is to relax r ∈ [0, ∞] in (3); see e.g., [26,13] for similar discussions.…”

“…for some small ε > 0 such that (13) and (14) hold. For J T,1 , note that in this case, it follows from (17) that |Υ r,̺ | = ρ(T )(1 − r(t − s))A(v, w), and by (2) that, we can choose small ε > 0 such that…”

“…

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.

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On the maxima of continuous and discrete time Gaussian order statistics processes

“…Tan and Hashorva (2014). Piterbarg's max-discretisation theorems have been extended to more general Gaussian cases, see Hüsler (2004), Hüsler and Piterbarg (2004), Tan and Hashorva (2014), Hashorva and Tan (2015) and Tan and Wang (2015). Although the Piterbarg's max-discretisation theorems for Gaussian processes have been studied extensively under different conditions in the past, it is far from complete.…”