Almost Sure and  \(L^p\) Convergence

This chapter is on convergence of martingales. We state and prove \(L^p\) and almost sure convergence results for martingales, sub-martingales and super-martingales. The crucial tool in developing these results is the Upcrossing lemma. It gives a bound for the expectation of the number of upcrossings of any interval by a sub-martingale, in terms of the expectation of the end point of the sequence and the end points of the interval. There are numerous results on the convergence of martingales, and we provide a sampling of these. The notion of reverse martingale (time-reversed martingales) is also introduced.