Selecting a sequence of last successes in independent trials

Let I1, I2, . . . , In be a sequence of independent indicator functions de- fined on a probability space (Ω, A, P ). We say that index k is a success time if Ik = 1. The sequence I1, I2, . . . , In is observed sequentially. The objective of this article is to predict the l-th last success, if any, with maximum probability at the time of its occurence. We find the optimal rule and discuss briefly an algorithm to compute it in an efficient way. This generalizes the result of Bruss (1998) for l = 1, and is equivalent to the problem of (multiple) stopping with l stops on the last l successes. We extend then the model to a larger class allowing for an unknown number N of indicator functions, and present, in particular, a convenient method for an approximate solution if the success probabilities are small. We also discuss some applications of the results.