A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory

Suppose that ( p n ) is an infinitely divisible distribution on the non-negative integers having Lévy measure ( v n ). In this paper we derive a necessary and sufficient condition for the existence of the limit lim n →∞ p n /v n . We also derive some other results on the asymptotic behaviour of the sequence ( P n ) and apply some of our results to the theory of fluctuations of random walks. We obtain a necessary and sufficient condition for the first positive ladder epoch to belong to the domain of attraction of a spectrally positive stable law with index α, α ∈ (1,2).