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 Ap + Bp, where up to a constant, Ap =V 2 - 1lp/(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 Ap. 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 Bp, generalizing another result of (Ann. Probab. 10 (1982) 672-688).