The Martingale Central Limit Theorem

One of the most useful generalizations of the central limit theorem is the martingale central limit theorem of Paul Lévy. Lévy was in part inspired by Lindeberg’s treatment of the central limit theorem for sums of independent – but not necessarily identically distributed – random variables. Lindeberg formulated what, in retrospect, is the right hypothesis, now known as the Lindeberg condition,1 on the summands for the central limit theorem, and in addition he proposed a new approach to proving central limit theorems. The Lindeberg condition plays a central role in the most general form of the martingale central limit theorem, as well, and as Lévy showed, the Lindeberg method of proof can be adapted to martingales. In this section I will show how Lindeberg’s method works in the very simplest context, for sums of independent, identically distributed random variables. In section 3, I will show how the method can be generalized to martingales.