A Geometric Approach for Bounding Average Stopping Time

We propose a geometric approach for bounding average stopping times defined in terms of sums of i.i.d. random variables. We consider stopping times in the hyperspace of sample number and sample sum. Our techniques relies on exploring geometric properties of continuity or stopping regions. Especially, we make use of the concepts of convex hull, convex sets and supporting hyperplane. Explicit formulae and efficiently computable bounds are obtained for average stopping times. Our techniques can be applied to bound average stopping times involving random vectors, nonlinear stopping boundary, and constraints of sample number. Moreover, we establish a stochastic characteristic of convex sets and generalize Jensen's inequality, Wald's equations and Lorden's inequality, which are useful for investigating average stopping times.