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

We consider a continuous, infinitely divisible random field in Rd 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.