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Cumulative shock models. (English) Zbl 0701.60084

The general setup in cumulative shock models is a family $\left\{\left({X}_{k},{Y}_{k}\right)$, $k\ge 0\right\}$ of i.i.d. two-dimensional random variables, where ${X}_{k}$ represents the magnitude of the k th shock and where ${Y}_{k}$ represents the time between the k th and the $\left(k+1\right)th$ shock. The system breaks down when the cumulative shock magnitude exceeds some given level. The object in focus is the lifetime of the system.

Now let $\left\{\left({W}_{k},{Z}_{k}\right)$, $k\ge 0\right\}$ be a sequence of i.i.d. random variables with E ${W}_{1}>0$ $\left({W}_{0}:={Z}_{0}:=0\right)$. Set

${U}_{n}:=\sum _{k=1}^{n}{W}_{k}\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}{V}_{n}:=\sum _{k=1}^{n}{Z}_{k},\phantom{\rule{1.em}{0ex}}n\ge 1,$

and define the first passage-time process $\left\{\tau$ (t), $t\ge 0\right\}$ by $\tau \left(t\right):=min\left\{n:{U}_{n}>t\right\}$. The random variable of interest is ${V}_{\tau \left(t\right)}$. With an appropriate choice of ${W}_{k}$ and ${Z}_{k}$ as functions of ${X}_{k}$ and ${Y}_{k}$ 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.

Reviewer: J.Tóth
##### MSC:
 60K05 Renewal theory 90B25 Reliability, availability, maintenance, inspection, etc. (optimization) 60K10 Applications of renewal theory