An extension of the renewal equation and its application in the collective theory of risk
An extension of the renewal equation and its application in the collective theory of risk
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DOI：
10.1080/03461238.1970.10405664
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发表时间：
1970-07
影响因子：
1.8
通讯作者：
H. Gerber
中科院分区：
文献类型：
--
作者：
H. Gerber
考虑更新方程，其中给出了(0，∞)上的z(X)和适当的概率分布F(X)。设µ=∫0∞x df(X)，则不排除µ=∞的情况。则下列定理等价于更新定理(见Feller[2])。定理1.1。如果z是直接Riemann可积的，且F不是算术的，则.缺陷更新方程具有重要的应用价值。这里L(X)是亏损概率分布，L∞<；1。我们有(见[2])定理1.2。如果存在z(∞)=LIMZ(X)，x→∞，则.
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 .