On the rate of convergence in the martingale central limit theorem

Details On the rate of convergence in the martingale central limit theorem On the rate of convergence in the martingale central limit theorem

2011-03-25

arxiv.org/abs/1103.5050v1

Mathematics

Probability

10 pages

Scientific paper

Consider a discrete-time martingale, and let V^2 be its normalized quadratic variation. As V^2 approaches 1 and provided some Lindeberg condition is satisfied, the distribution of the rescaled martingale approaches the Gaussian distribution. For any p at least equal to 1, E. Haeusler (1988) gives 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 A_p is, up to a constant, the L^p norm of V^2-1 to the power p/(2p+1). We discuss here the optimality of this term, focusing on the restricted class of martingales with bounded increments. In this context, E. Bolthausen (1982) sketches a strategy to prove optimality for p = 1. Here, we extend this strategy to any p > 1, thus 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 E. Bolthausen (1982).

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