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(∞),
bilinear, Volterra processes, ..., enter this frame.