Home OALib Journal OALib PrePrints Submit Ranking News My Lib FAQ About Us Follow Us+
 All Title Author Keywords Abstract
 Publish in OALib Journal ISSN: 2333-9721 APC: Only \$99

 Relative Articles On the rate of convergence in the central limit theorem for martingale difference sequences On the central and local limit theorem for martingale difference sequences Uniform Convergence and the Central Limit Theorem Conditional central limit theorem via martingale approximation A central limit theorem for fields of martingale differences On the functional central limit theorem via martingale approximation On the rate of convergence and Berry-Esseen type theorems for a multivariate free central limit theorem Moment and tail estimates for martingales and martingale transform, with application to the martingale limit theorem in Banach spaces Exactness of martingale approximation and the central limit theorem Exact convergence rates in the central limit theorem for a class of martingales More...
Statistics  2011

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

 Full-Text   Cite this paper

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).

Full-Text