On maxima of stationary fields

Abstract

Let {Xn : n ∈ Z d } be a weakly dependent stationary random field with maxima MA := sup{X i : i ∈ A} for finite A ⊂ Z d and Mn := sup{X i : 1 ≤ i ≤ n} for n ∈ N d . In a general setting we prove that P(M (N 1 (n),N 2 (n),...,N d (n)) ≤ vn) = exp(−n d P(X0 > vn, MA n ≤ vn)) + o(1) for some increasing sequence of sets An of size o(n d ), where (N1(n), N2(n), . . . , N d (n)) → (∞, ∞, . . . , ∞) and N1(n)N2(n) · · · N d (n) ∼ n d . The sets An are determined by a translation invariant total order on Z d . For a class of fields satisfying a local mixing condition, including m-dependent ones, the main theorem holds with a constant finite A replacing An. The above results lead to new formulas for the extremal index for random fields. The new method of calculating limiting probabilities for maxima is compared with some known results and applied to the moving maximum field.

Extremal clustering and cluster counting for spatial random fields

“…[8] for d = 1. This result generalizes the claims in [1,12], dealing with the one-dimensional case and the case of index sets being boxes, respectively, to the present framework allowing for a much more involved asymptotic development of the index sets.…”

“…In the literature, results for spatial objects, comparable to some of the present results, are to the best of the authors' knowledge only formulated under the assumption of (D n ) n∈N being a sequence of increasing boxes; see for instance [5,12] that provides results like the ones found in Section 3 below under this additional geometrical assumption. In contrast, we allow the index sets D n to expand in a much more general way; we refer to the authors' papers [10,13,14] for similar however slightly less general assumptions on the sequence of index sets, but with other and less general results.…”

Extremes of Lévy-driven spatial random fields with regularly varying Lévy measure

“…Since the cube C L (0) ⊆ R d is a convex body and g α is decreasing, we find by Lemma 6, equation (26), and the homogeneity of the intrinsic volumes that there are constants µ j independent of C L (0) such that…”

“…The second part of the paper concerns the asymptotic distribution of sup v∈C n X v as n → ∞, where (C n ) is a sequence of index sets in R d increasing appropriately. From extreme value theory for dependent stationary fields we know, assuming some mixing and anti-clustering conditions, that the distribution of the running maximum of a stationary field is determined by its marginal tail; see [10,14,18] for detailed treatments of classical extreme value theory, and see [12,26,27] for generalizations to stationary, discretely indexed d-dimensional spatial fields. In particular, if the marginal tail is in the maximum domain of attraction of the Fréchet distribution (or equivalently it is regularly varying), then the running maximum of the field converges to the Fréchet distribution.…”

On extremal index of max-stable stationary pro­cesses

“…We note that θ X given in (5.8) is the extremal index of a large class of stationary rf's with representation (5.7), see e.g., [4,46]. The main assumption on V i 's in the aforementioned references is that they are independent, with identical distribution and regularly varying at infinity with index α > 0, whereas for c i 's it is further assumed that t∈Z d c β t < ∞ for some β ∈ (0, α).…”