On the rate of convergence in the martingale central limit theorem

Abstract

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 $p\geq 1$, (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, $A_{p}={\|V^{2}-1\|}_{p}^{p/(2p+1)}$. 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 $p\geq 1$, 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).