The author studies system lifetimes that are associated with a cumulative shock model as follows: The system fails when the cumulative damage exceeds some threshold, or when a single large shock occurs. H. Li
and the reviewer [Stochastic Processes Appl. 58, No. 2, 205-216 (1995; Zbl 0833.60088
); Appl. Probab. 34, No. 1, 1-13 (1997; Zbl 0952.60085
)] obtained some aging properties of such lifetimes. In the present paper the author obtains the asymptotic distribution of such lifetimes. Some uniform integrability and moment convergence results, as well as some applications and further results are also included in the paper.