We derive a central limit theorem for triangular arrays of possibly nonstationary random variables satisfying a condition of weak dependence in the sense of Doukhan and Louhichi [Stoch. Proc. Appl. 84 (1999) 313–342]. The proof uses a new variant of the Lindeberg method: the behavior of the partial sums is compared to that of partial sums of dependent Gaussian random variables. We also discuss a few applications in statistics which show that our central limit theorem is tailor-made for statistics of different type.

In this paper, a very useful lemma (in two versions) is proved: itsimplifies notably the essential step to establish a Lindebergcentral limit theorem for dependent processes. Then, applying thislemma to weakly dependent processes introduced in Doukhan andLouhichi (1999), a new central limit theorem is obtained forsample mean or kernel density estimator. Moreover, by using thesubsampling, extensions under weaker assumptions of these centrallimit theorems are provided. All the usual causal or non causaltime series: Gaussian, associated, linear, ARCH(∞),bilinear, Volterra processes, ..., enter this frame.