An extension of the renewal equation and its application in the collective theory of risk
- 1 July 1970
- journal article
- research article
- Published by Taylor & Francis in Scandinavian Actuarial Journal
- Vol. 1970 (3-4) , 205-210
- https://doi.org/10.1080/03461238.1970.10405664
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 .Keywords
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