The general setup in cumulative shock models is a family , of i.i.d. two-dimensional random variables, where represents the magnitude of the k th shock and where represents the time between the k th and the 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 , be a sequence of i.i.d. random variables with E . Set
and define the first passage-time process (t), by . The random variable of interest is . With an appropriate choice of and as functions of and 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.