Tail asymptotics for the supremum of an infinitely divisible field with convolution equivalent Lévy measure

Abstract

We consider a continuous, infinitely divisible random field in R d given as an integral of a kernel function with respect to a Lévy basis with convolution equivalent Lévy measure. For a large class of such random fields we compute the asymptotic probability that the supremum of the field exceeds the level x as x → ∞. Our main result is that the asymptotic probability is equivalent to the right tail of the underlying Lévy measure.

Excursion sets of infinitely divisible random fields with convolution equivalent Lévy measure

“…As argued in [13], all the fields U n , X 1 , and X 2 have continuous extension to B ⊕C r . It should furthermore be noted that each of the fields (U n t ) t∈B⊕Cr can be represented by…”

“…We shall make the same general assumptions as in [13] except for the additional assumption (11) below. For completeness, we will present all assumptions in the following.…”

“…Alternatively, this can be expressed as For the study of the extremal behaviour of (X t ) t∈B⊕Cr , it will crucial that the field (X t ) t∈T is itself infinitely divisible, with T = (B ⊕ C r ) ∩ Q d , where Q d are the rational numbers in R d . For details, see [13] and references therein. The Lévy measure of (X t ) t∈T is the measure ν on (…”

“…We will assume that the Lévy measure of the random measure M has a convolution equivalent right tail ( [5,6,10]). In [13] it was shown under some regularity conditions that the distribution of sup t∈B X t has a similar convolution equivalent tail. In the present paper we will be interested in the excursion set A x = {t : X t > x} .…”

“…The proof will be in several steps, utilising that X can be decomposed as X 1 + X 2 , where X 1 is a compound Poisson sum and X 2 has lighter tails than X 1 . The proofs in this section will apply techniques that are similar to the proofs in [13].…”