First, we consider a stationary random field indexed by an increasing sequence of subsets of \(\mathbb {Z}^{d}\). Under certain mixing and anti–clustering conditions combined with a very broad assumption on how the sequence of spatial index sets increases, we obtain an extremal result that relates a normalized version of the distribution of the maximum of the field over the index sets to the tail distribution of the individual variables. Furthermore, we identify the limiting distribution as an extreme value distribution. Secondly, we consider a continuous, infinitely divisible random field indexed by \(\mathbb {R}^{d}\) given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. When observing the supremum of this field over an increasing sequence of (continuous) index sets, we obtain an extreme value theorem for the distribution of this supremum. The proof relies on discretization and a conditional version of the technique applied in the first part of the paper, as we condition on the high activity and light–tailed part of the field.

首先，我们考虑由\（\ mathbb {Z} ^ {d} \）的子集的增加序列索引的固定随机场。在一定的混合和反聚簇条件下，结合关于空间索引集序列如何增加的非常宽泛的假设，我们获得了一个极值结果，该结果将索引集上最大场分布的归一化版本与尾部相关联各个变量的分布。此外，我们将极限分布确定为极值分布。其次，我们考虑由\（\ mathbb {R} ^ {d} \）索引的连续，无限可分的随机字段以卷积等效的Lévy度量作为Lévy基的内核函数的积分给出。当在（连续）索引集的递增序列上观察该字段的最大值时，我们获得该最大值的分布的极值定理。证明依赖于离散化和在本文的第一部分中应用的该技术的条件版本，因为我们以该领域的高活动性和轻尾部分为条件。