TY - JOUR

T1 - On the rate of convergence in the martingale central limit theorem

AU - Mourrat, Jean Christophe

N1 - Copyright:
Copyright 2013 Elsevier B.V., All rights reserved.

PY - 2013/5

Y1 - 2013/5

N2 - 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).

AB - 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).

KW - Central limit theorem

KW - Martingale

KW - Rate of convergence

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U2 - 10.3150/12-BEJ417

DO - 10.3150/12-BEJ417

M3 - Article

AN - SCOPUS:84884149278

VL - 19

SP - 633

EP - 645

JO - Bernoulli

JF - Bernoulli

SN - 1350-7265

IS - 2

ER -