Probability of ruin in a discrete-time risk model

We continue the study of the discrete-time risk model introduced by Picard et al. (2003). The cumulative loss process (S-t)(tis an element ofN) has independent and stationary increments, the increments per unit of time having nonnegative integer values with distribution {a(i), i is an element of N} and mean (a) over bar. The premium receipt process (c(k))(kis an element ofN) is deterministic, nonnegative and nonuniform; in addition, we assume it to be regular in order for there is bounded as the time to exist a constant c > a such that the deviation Sigma(k=0)(t) (c(k) - c) t varies. We are interested in whether or not ruin occurs within a finite time. If T is the time of ruin, we obtain P(T = infinity) as the limit of P(T > t) as t --> infinity, firstly in the particular case where c = 1/d for some positive d E N, and then in the general case for positive c under the condition that a(0) > 1/2.