Infinitely divisible sequences

Abstract

Sequences and related by the system of equations occur frequently in a number of areas of mathematics, and when they do one is often interested in relating the asymptotic behaviours of the two sequences. In probability theory sequences _n , which arise when one has the added conditions b₀ > 0 and a_j⩾ 0, often occur. In this paper we characterize the class of sequences _n that can arise in this way. We also examine the asymptotic behaviour of these sequences and prove, in particular, a sufficient condition for the convergence of n to a non zero limit. Our results admit interpretations in a number of fields. For example renewal sequences fit this pattern and the renewal theorem with finite mean recurrence time is a special case of our convergence theorem.