Martingale Methods

We survey the ways that martingales and the method of gambling teams can be used to obtain otherwise hard-to-get information for the moments and distributions of waiting times for the occurrence of simple or compound patterns in an independent or a Markov sequence. We also survey how such methods can be used to provide moments and distribution approximations for a variety of scan statistics, including variable length scan statistics. Each of the general problems considered here is accompanied by one or more concrete examples that illustrate the computational tractability of the methods.