Open Access
May 2013 On the rate of convergence in the martingale central limit theorem
Jean-Christophe Mourrat
Bernoulli 19(2): 633-645 (May 2013). DOI: 10.3150/12-BEJ417


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).


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Jean-Christophe Mourrat. "On the rate of convergence in the martingale central limit theorem." Bernoulli 19 (2) 633 - 645, May 2013.


Published: May 2013
First available in Project Euclid: 13 March 2013

zbMATH: 1277.60051
MathSciNet: MR3037167
Digital Object Identifier: 10.3150/12-BEJ417

Keywords: central limit theorem , martingale , rate of convergence

Rights: Copyright © 2013 Bernoulli Society for Mathematical Statistics and Probability

Vol.19 • No. 2 • May 2013
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