3 - The Central Limit Theorem

Abstract

This chapter focuses on the central limit theorem. A large variety of negligibility assumptions have been made about the differences Xni during the formulation of martingale central limit theorems. The classic condition of negligibility in the theory of sums of independent r.v. requires Xni to be uniformly asymptotically negligible. This is generally a little weaker than the summation condition. The conditional variance Vn2 is one of several estimates of the variance ESn2. It is an intrinsic measure of time for the martingale. For many purposes, the time taken for a martingale to cross a level is best represented through its conditional variance rather than the number of increments up to the crossing. The duality between the definitions of forward and reverse martingales suggests that forward martingale limit theorems should have reverse martingale duals. In the case of the central limit theorem, the analog is perhaps best presented by considering infinite martingale arrays.