Extremes of Gaussian random fields with regularly varying dependence structure

Abstract

Let X(t), t ∈ T be a centered Gaussian random field with variance function σ 2 (·) that attains its maximum at the unique point t 0 ∈ T , and let M (T ) := sup t∈T X(t). For T a compact subset of R, the current literature explains the asymptotic tail behaviour of M (T ) under some regularity conditions including that 1 − σ(t) has a polynomial decrease to 0 as t → t 0 . In this contribution we consider more general case that 1 − σ(t) is regularly varying at t 0 . We extend our analysis to random fields defined on some compact T ⊂ R 2 , deriving the exact tail asymptotics of M (T ) for the class of Gaussian random fields with variance and correlation functions being regularly varying at t 0 . A crucial novel element is the analysis of families of Gaussian random fields that do not possess locally additive dependence structures, which leads to qualitatively new types of asymptotics.

On maximum of Gaussian random field having unique maximum point of its variance

“…In [2] the covariance function satisfying 1 − r(t) = ρ 2 1 (|a 11 t 1 + a 12 t 2 |) + ρ 2 2 (|a 21 t 1 + a 22 t 2 |), t → 0, is considered, where ρ i , i = 1, 2, are regularly varying at zero functions with indexes α i ∈ (0, 1]. The rank of matrix A = (a ij , i, j = 1, 2) can be 2, 1, or 0.…”

“…Here such degenerated cases are not considered, but in a corresponding basis one can represent R d as a product of two spaces, dimension of one of them should be equal to the rank of a matrix which generalized A to d-dimension case, with subsequent application of given here results. Notice that in [2] the function 1 − σ(t) is also assumed to have a similar to 1 − r(t) form, with some other regularly varying functions. The corresponding matrix must be, of course, not degenerated, otherwise one has infinitely many maximum points.…”

“…From results here it follows that such restriction on the variance is not necessary, in contrast of the above representation for the covariance function. We would like to mention that discussions with authors of [2] helped us a lot in formulation of our Conditions.…”

“…Condition 4 There exists a basis in R d , a vector function q(u) = (q 1 (u), ..., q d (u)), u > 0, q i (u) > 0, i = 1, ..., d, and a positive for all t = 0 function h(t) such that for any t written in these coordinates, lim u→∞ u 2 (1 − r(q(u)t)) =h(t), (2) uniformly in t from any closed set.…”

Approximation of Sojourn Times of Gaussian Processes

“…The existence of such a Gaussian process is guaranteed by the Assertion in [32][p.265] and follows from [33,34]. Consequently, by Slepian lemma and [30][Lemma 5.1] for any η ≥ 0 and sufficiently large u…”