Let , , , be i.i.d. nonnegative random vectors and let , , and , for a fixed subset of . Interpretation: a system subject to load cycles or shocks with magnitudes and durations or intershock times . Then is the number of the shock where for the first time successive shocks have magnitudes in a critical region . The total duration or time up to this shock is . Another interpretation in insurance: claims and interclaim times.
The paper derives the Laplace-Stieltjes transform of , the probability generating function of and the distribution function of by means of recurrence w.r. to . The first moment of , in terms of and , and its variance are derived. A condition on as ensures the asymptotic behaviour in distribution of as for fixed . A similar one is derived for fixed and small such that . The conditions imply a form of regular variation. These derivations use only the recurrence for .