Rates of Convergence in the CLT for Some Weakly Dependent Random Variables

We provide rates in the central limit theorem (CLT) for some weakly dependent sequences under a power decay of their covariance. Those sequences are assumed to be associated with or to satisfy a common property of Gaussian processes and positively (or negatively) dependent random variables. For this, we extend the Lindeberg method in our framework, following a method due to [E. Rio, Probab. Theory Related Fields, 104 (1996), pp. 255–282] The method of the proofs also provides upper bounds of the Dudley distances between the distribution of a normalized sum of those weak dependent random variables and the standard normal distribution. It also leads to Rosenthal-type inequalities for moments of partial sums. for mixing sequences.