A simple compound scan statistic useful for modeling insurance and risk management problems

Let us consider a sequence of binary success/failure trials and denote by Tk the waiting time for the first occurrence of two successes separated by at most k failures, where k≥0 is any integer. Let also Y1,Y2,… be a sequence of independent and identically distributed (i.i.d) discrete random variables, independent of Tk. In the present article we develop some results for the distribution of the compound random variable Sk=∑t=1TkYt and illustrate how these results can be profitably used to study models pertaining to actuarial science and financial risk management practice.