1 - Stopping times

Summary

We will begin with the material that will be used throughout the book. The idea of the stopping time, especially the simple stopping time, is central. The setting in which this naturally occurs involves Moore-Smith convergence, or convergence of nets or generalized sequences. This will be useful even if we are interested only in sequences of real-valued random variables; but will be even more useful when we consider derivation (Chapter 7) and processes indexed by directed sets (Chapter 4).

Given a stochastic process (X_n), a stopping time is a random variable τ taking values in IN ∪ {∞} such that, for each k, the event {τ = k} is determined by the first k random variables X₁, X₂, …, X_k A process (X_n) is an amart iff for every increasing sequence τ_n of bounded stopping times, E[X_τn] converges. (For variants of this definition, see Section 1.2.) The main result of this chapter is the amart convergence theorem for the index set IN, proved in Section 1.2. The argument, using stopping times, is elementary, and may be followed by a reader with only a basic knowledge of the measure theory. To make the point, we will sketch the proof of almost sure convergence of an amart (X_n) with integrable supremum. The basic observation is that there is an increasing sequence τ_n of simple stopping times such that X_τn converges in probability to X* = lim sup X_n. The reason for this is that lim sup or any other accumulation point manifests itself infinitely often on the way to infinity; it is like a light shining on the horizon.