Necessary and sufficient conditions for the conditional central limit theorem

• 成果类型: Article
• 署名作者: Dedecker, J; Merlevède, F
• 署名单位: Sorbonne Universite
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 1044-1081
• 关键词: invariance-principles; stationary-processes; subgeometric rates; mixing sequences; CONVERGENCE; dependence

摘要

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 Renyi) 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.