Extreme value theory for spatial random fields – with application to a Lévy-driven field

Extremes for stationary regularly varying random fields over arbitrary index sets

“…Surprisingly, the shape of the asymptotic local region Υ is arbitrary. In contrast with existing results such as Stehr and Rónn-Nielsen [21] we show that convexity is not required when dealing with extremes. Under the anti-clustering condition, we deal with index sets that are local regions such as Υ reproduced over a lattice.…”

“…The framework of [21,22] is a particular specification of our framework. Indeed, consider Assumption 1 in [21] (which is Assumption 3 in [22]): The sequence (C n ) n∈N consists of p-convex bodies (i.e. connected sets which are also unions of p convex sets), where…”

“…For any convex body C, we have that V 0 (C) = 1 and V 1 (C) is equal to the perimeter of C divided by π. Then, Assumption 1 in [21] states that the sum of the perimeters of the C n,i s must not grow faster than the square root of the volume of C n . There are cases where this is not true, like when one of the C n,i s is a rectangle with edges increasing with different speed.…”

“…Let us consider Λ n as an arbitrary index set of Z k \ {0}. Such an extension of the rectangular index set is not straightforward, as Stehr and Rónn-Nielsen [21] showed in the asymptotically independent case (Θ t = 0 for all t = 0). For index sets (Λ n ), a geometric condition must hold.…”

“…Under the so-called anti-clustering condition, the extremal dependence is only local. Thus the index set can have a general form compared to previous literature [3,21]. However, we cannot describe the clustering of extreme values in terms of the usual spectral tail measure [23] except for hyperrectangles or index sets in the lattice case.…”