Tail asymptotics of an infinitely divisible space-time model with convolution equivalent Lévy measure

Abstract

We consider a space-time random field on ${{\mathbb{R}^d} \times {\mathbb{R}}}$ given as an integral of a kernel function with respect to a Lévy basis with a convolution equivalent Lévy measure. The field obeys causality in time and is thereby not continuous along the time axis. For a large class of such random fields we study the tail behaviour of certain functionals of the field. It turns out that the tail is asymptotically equivalent to the right tail of the underlying Lévy measure. Particular examples are the asymptotic probability that there is a time point and a rotation of a spatial object with fixed radius, in which the field exceeds the level x, and that there is a time interval and a rotation of a spatial object with fixed radius, in which the average of the field exceeds the level x.

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

“…for all ǫ > 0 and all L ∈ N. Let T ⊂ C L (0) be the countable separating dense subset associated to the separable field (Y C v ) v∈C L (0) . By considerations as in [19] and [28], the countable field (Y C v ) v∈T is infinitely divisible with a characteristic function given as in [6, Eq. (1.1)].…”

“…Lemma 15. Let (Z (t) v ) v and (Y v ) v be given by (28) and (20), respectively. For all 0 < t < ∞ there is a sequence of functions g L satisfying lim sup…”

“…where |B| is Lebesgue measure of B ⊆ R d and K is a computable constant. A slightly more general result is given in [28], in which a space-time Lévy model X v,t with a convolution equivalent Lévy measure is considered. For a large class of functionals Ψ acting on the field, it is shown that there are constants c, C such that…”