Rates in the Empirical Central Limit Theorem for Stationary Weakly Dependent Random Fields

Abstract

A weak dependence condition is derived as the natural generalization to random fields on notions developed in Doukhan and Louhichi (1999). Examples of such weakly dependent fields are defined. In the context of a weak dependence coefficient series with arithmetic or geometric decay, we give explicit bounds in Prohorov metric for the convergence in the empirical central limit theorem. For random fields indexed by &Zopf d , in the geometric decay case, rates have the form n −1/(8d+24) L(n), where L(n) is a power of log(n).