On the rate of convergence in the martingale central limit theorem

Jean Christophe Mourrat

Research output: Contribution to journalArticle

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 ≥ 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).

Original languageEnglish (US)
Pages (from-to)633-645
Number of pages13
JournalBernoulli
Volume19
Issue number2
DOIs
StatePublished - May 2013

Keywords

  • Central limit theorem
  • Martingale
  • Rate of convergence

ASJC Scopus subject areas

  • Statistics and Probability

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