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

Abstract

Motivated by the papers of Piterbarg [17] and Hüsler [7], in this paper the asymptotic relation between the maximum of a continuous dependent homogeneous Gaussian random field and the maximum of this field sampled at discrete time points is studied. It is shown that, for the weakly dependent case, these two maxima are asymptotically independent, dependent or coincide when the grid of the discrete time points is a sparse grid, Pickands grid or dense grid, respectively, while for the strongly dependent case, these two maxima are asymptotically totally dependent if the grid of the discrete time points is sufficiently dense, and asymptotically dependent if the grid points are sparse or Pickands grids.

Maxima and minima of homogeneous Gaussian random fields over continuous time and uniform grids

“…For the sparse grids, let Proof. The proof is similar to Lemma 1 of Piterbarg (2004) and Lemma A1 of Tan and Wang (2015).…”

“…The grid is called a Pickands grid if all D i ∈ (0, ∞). Under conditions A1 − A3, Tan and Wang (2015) derived the limiting distribution of M T when the uniform grid is sparse grid, Pickands' grid and dense grid, respectively.…”

“…α d /2 (•) are independent fractional Brownian motions, cf. Piterbarg (2004) and Tan and Wang (2015). Further, let the bivariate normalizing constants u T and v T be given by…”

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.…”

An extension of almost sure central limit theorem for the maximum of stationary Gaussian random fields

“…Further, Choi (2010), Tan and Wang (2014), Zang (2014) and Tan and Wang (2015) studied the maximum of Gaussian random fields and obtained its almost sure limit theorem. Random field theory has recently received increasing attention.…”