  In this paper, a very useful lemma (in two versions) is proved: it simplifies
notably the essential step to establish a Lindeberg central limit theorem for
dependent processes. Then, applying this lemma to weakly dependent processes
introduced in Doukhan and Louhichi (1999), a new central limit theorem is
obtained for sample mean or kernel density estimator. Moreover, by using the
subsampling, extensions under weaker assumptions of these central limit
theorems are provided. All the usual causal or non causal time series:
Gaussian, associated, linear, ARCH($\infty$), bilinear, Volterra
processes,$...$, enter this frame.
