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Zbl 0701.60084
Gut, Allan
Cumulative shock models.
(English)
[J] Adv. Appl. Probab. 22, No.2, 504-507 (1990). ISSN 0001-8678

The general setup in cumulative shock models is a family $\{(X\sb k,Y\sb k)$, $k\ge 0\}$ of i.i.d. two-dimensional random variables, where $X\sb k$ represents the magnitude of the k th shock and where $Y\sb k$ represents the time between the k th and the $(k+1) th$ shock. The system breaks down when the cumulative shock magnitude exceeds some given level. The object in focus is the lifetime of the system. \par Now let $\{(W\sb k,Z\sb k)$, $k\ge 0\}$ be a sequence of i.i.d. random variables with E $W\sb 1>0$ $(W\sb 0:=Z\sb 0:=0)$. Set $$U\sb n:=\sum\sp{n}\sb{k=1}W\sb k\text{ and } V\sb n:=\sum\sp{n}\sb{k=1}Z\sb k,\quad n\ge 1,$$ and define the first passage-time process $\{\tau$ (t), $t\ge 0\}$ by $\tau (t):=\min \{n: U\sb n>t\}$. The random variable of interest is $V\sb{\tau (t)}$. With an appropriate choice of $W\sb k$ and $Z\sb k$ as functions of $X\sb k$ and $Y\sb k$ it is shown how to apply all the previous results of the author on stopped random walks (such as the strong law, the central limit theorem, and the law of iterated logarithm) to shock models.
[J.Tóth]
MSC 2000:
*60K05 Renewal theory
90B25 Reliability, etc.
60K10 Appl. of renewal theory

Keywords: renewal theory; insurance risk; cumulative shock models; cumulative shock magnitude; stopped random walks; central limit theorem

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