Extreme Shock Models

The standard assumptions in shock models are that the failure (of the system) is related either to the cumulative effect of a (large) number of shocks or that failure is caused by a shock which is larger than a certain critical level. The present paper is devoted to the second kind. Here the standard setting is that the shocks X_k, k 1, and the times between the shocks Y_k, k 1, are independent, identically distributed random vectors (X_k, Y_k), k 1. In particular, X_k and Y_k may well be dependent (the typical case). The main object of interest is the time to failure, T_(t), where T_n = _kn Y_k and (t) is the first exceedance time, viz. the first time that X_k > t. We derive moment relations and asymptotic distributions of T_(t) as t increases in such a way that P{X₁} > t} tends to 0. A final section discusses some extensions; more general events of failure, the non-i.i.d. case, and point process convergence for a particular case.