Running minimum in the best-choice problem

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 $$X_1,\cdots ,X_n$$X 1 , ⋯ , X n of i.i.d. random variables with continuous distribution. In this paper we look at more general models, where independent $$X_j$$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 $$\{j,\cdots ,n\}$${ j , ⋯ , n } , and in the second it is uniform on $$\{1,\cdots , n\}$${ 1 , ⋯ , n } .