Necessary and sufficient conditions for the conditional central limit theorem

Abstract

Following Lindeberg's approach, we obtain a new condition for a stationary sequence of square-integrable and real-valued random variables to satisfy the central limit theorem. In the adapted case, this condition is weaker than any projective criterion derived from Gordin's theorem [Dokl. Akad. Nauk SSSR 188 (1969) 739-741] about approximating martingales. Moreover, our criterion is equivalent to the conditional central limit theorem, which implies stable convergence (in the sense of Rényi) to a mixture of normal distributions. We also establish functional and triangular versions of this theorem. From these general results, we derive sufficient conditions which are easier to verify and may be compared to other results in the literature. To be complete, we present an application to kernel density estimators for some classes of discrete time processes.