Asymptotic distributions of maxima of complete and incomplete samples from multivariate stationary Gaussian sequences

Abstract

a b s t r a c tLet (X n ) be a sequence of d-dimensional stationary Gaussian vectors, and let M n denote the partial maxima of {X k , 1 ≤ k ≤ n}. Suppose that there are missing data in each component of X k and let M n denote the partial maxima of the observed variables. In this note, we study two kinds of asymptotic distributions of the random vector ( M n , M n ) where the correlation and cross-correlation satisfy some dependence conditions.

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

“…iii) We proceed as for the proof of cases i) and ii) using the bound (23). By Lemmas 2 and 3 in [27] and (19), (20) we obtain…”

“…b) The Berman condition is relaxed by assuming that (3) holds for some r ∈ [0, ∞). When r > 0 the Gaussian process X is said to be strongly dependent, see [22,26,23,32,29,8] for details on the extremes of such Gaussian processes. The contribution [34] derives Piterbarg's max-discretisation theorem for strongly dependent Gaussian processes.…”

On Piterbarg theorem for maxima of stationary Gaussian sequences

“…Furthermore, we generalize the recent findings of [16] which are motivated by [6]. For some related work on asymptotic behavior of extremes of Gaussian sequences see [2,11]. Brief organization of the rest of the paper: Section 2 presents the main results, their proofs are relegated to Section 3.…”

The Limit Properties of Maxima of Stationary Gaussian Sequences Subject to Random Replacing

“…The result in (3) has been extended to many other cases; we refer to [4,5] for Gaussian cases; ref. [6,7] for the almost sure limit theorem; ref.…”