An extension of the renewal equation and its application in the collective theory of risk

Abstract Let us consider the renewal equation where z(x) and the proper probability distribution F(x) on (0,∞) are given. Let µ = ∫0 ∞ x dF(x), the case µ = ∞ is not excluded. Then the following theorem is equivalent to the renewal theorem (see Feller [2]). Theorem 1.1. If z is directly Riemann integrable and F is not arithmetic, then . The defective renewal equation is of great importance for applications. There L(x) is a defective probability distribution, L ∞ < 1. We have (see [2]) Theorem 1.2. If z(∞) = lim z(x), x → ∞, exists, then .