On systems of renewal equations

Abstract

so the results contained in this paper are generalizations of facts from ordinary renewal theory. Our principal result is Theorem 2.2 which concerns the behavior of the functions M,(t) as t + co. Part (i) of this theorem generalizes Blackwell’s theorem and part (ii) is a generalization of the key renewal theorem. The study of the system of equations (1 .l) was motivated by a problem concerning the moments of a multi-dimensional age-dependent branching process. In Section 4 we shall apply our theory to this problem. The analytic techniques employed by Feller [2] in his treatment of renewal theory will be used extensively along with some facts from the Frobenius theory of positive matrices. It should be mentioned that Bellman and Cooke [l] have suggested another way to determine the asymptotic behavior of the functions M<(t) that utilizes either the theory of poles and residues or Tauberian theorems.