On the rate of convergence in the martingale central limit theorem

Consider a discrete-time martingale, and let V-2 be its normalized quadratic variation. As V-2 approaches 1, and provided that some Lindeberg condition is satisfied, the distribution of the rescaled martingale approaches the Gaussian distribution. For any p >= 1, (Ann. Probab. 16 (1988) 275-299) gave a bound on the rate of convergence in this central limit theorem that is the sum of two terms, say A(p) + B-p, where up to a constant, A(p) = parallel to V-2 - 1 parallel to(p/(2p+1))(p). Here we discuss the optimality of this term, focusing on the restricted class of martingales with bounded increments. In this context, (Ann. Probab. 10 (1982) 672-688) sketched a strategy to prove optimality for p = 1. Here we extend this strategy to any p >= 1, thereby justifying the optimality of the term A(p). As a necessary step, we also provide a new bound on the rate of convergence in the central limit theorem for martingales with bounded increments that improves on the term B-p, generalizing another result of (Ann. Probab. 10 (1982) 672-688).