Running minimum in the best-choice problem

Abstract

The full-information best choice problem asks one to find a strategy maximising the probability of stopping at the minimum (or maximum) of a sequence X_1,cdots ,X_n of i.i.d. random variables with continuous distribution. In this paper we look at more general models, where independent X_j’s may have different distributions, discrete or continuous. A central role in our study is played by the running minimum process, which we first employ to re-visit the classic problem and its limit Poisson counterpart. The approach is further applied to two explicitly solvable models: in the first the distribution of the jth variable is uniform on {j,cdots ,n}, and in the second it is uniform on {1,cdots , n}.