On the rate of convergence in the martingale central limit theorem

TL;DR

This paper investigates the rate of convergence in the martingale central limit theorem, establishing the optimality of certain bounds and providing new bounds for martingales with bounded increments.

Contribution

It extends the proof of optimality for the convergence rate bound to all p ≥ 1 and introduces improved bounds for martingales with bounded increments.

Findings

Confirmed the optimality of the rate term A_p for all p ≥ 1.

Provided a new, improved bound on the convergence rate B_p for bounded increment martingales.

Generalized previous results to a broader class of martingales with bounded increments.

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).