This chapter dives into the theory of (discrete time) martingales.The optional stopping theorem and the martingale convergence theorem are proved.These are used to provide some initial results regarding random walks on groups and bounded harmonic functions. Specifically, the random walk on the integer line is shown to be recurrent. Also, it is shown that the space of bounded harmonic functions is either just the constant functions or has infinite dimension.

In this chapter, we turn to martingales, which play a central role in probability theory. We illustrate their use in a number of applications to the analysis of discrete stochastic processes. After some background on stopping times and a brief review of basic martingale properties and results, we develop two major directions. We show how martingales can be used to derive a substantial generalization of our previous concentration inequalities – from the sums of independent random variables we focused on previously to nonlinear functions with Lipschitz properties. In particular, we give several applications of the method of bounded differences to random graphs. We also discuss bandit problems in machine learning. In the second thread, we give an introduction to potential theory and electrical network theory for Markov chains. This toolkit in particular provides bounds on hitting times for random walks on networks, with important implications in the study of recurrence among other applications. We also introduce Wilson’s remarkable method for generating uniform spanning trees.