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

Abstract

Suppose that (/>") is an infinitely divisible distribution on the non-negative integers having Levy measure (v n ). In this paper we derive a necessary and sufficient condition for the existence of the limit lim ; , -ooPn/"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 a, a G (1,2).1980 Mathematics subject classification (Amer. Math. Soc): 670 E 07, 60 J 15.

Asymptotics for the First Passage Times of Lévy Processes and Random Walks

“…increments and assume that S n → −∞ a.s. Then an analogue of τ x is the stopping time ν x = min{n ≥ 1 : S n < −x}. Asymptotics for ν x have been studied in [12] for x = 0, and in [10], [6] for x > 0. In [12], it is shown that P{ν 0 > n} ∼ P{S n ≥ 0}/n if the latter is a subexponential sequence (see definition below).…”

“…Asymptotics for ν x have been studied in [12] for x = 0, and in [10], [6] for x > 0. In [12], it is shown that P{ν 0 > n} ∼ P{S n ≥ 0}/n if the latter is a subexponential sequence (see definition below). In [10] and [6], the asymptotics for ν x have been found when x > 0.…”

Transition phenomena for ladder epochs of random walks with small negative drift

“…The situation when Eτ < ∞, which is a particular case of the relative stability, was considered by Embrechts and Hawkes [5]. There it has been shown that If the expectation EX is finite, then the condition ∞ k=1 k −1 P(S k ≤ 0) = ∞ is equivalent to the inequality EX ≤ 0, see again [19,Theorem 17.1].…”

Generalized Logarithmic Series Distribution

“…:In (nPn )-1--+ 00 as n --+ 00 • As in Embrechts and Hawkes (1982), we have that LC n is absolutely convergent, LC n =0 and c n ---qn ,(n _(0). It follows from (3.3) and (3.4) that…”

“…We will therefore concentrate on asymptotic expressions for CY n as 11 _ 00. The following theorem of Embrechts and Hawkes (1982) illustrates that such expressions are closely related to the convolution behavior of Pn as n _ 00.…”