Limit theorems for functionals of moving averages

• 成果类型: Article
• 署名作者: Ho, HC; Hsing, T
• 署名单位: Academia Sinica - Taiwan; Texas A&M University System; Texas A&M University College Station
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1023481106
• 发表日期: 1997
• 页码: 1636-1669
• 关键词: mixing properties; random-variables; gaussian fields; CONVERGENCE; memory

摘要

Let X-n = Sigma(i=1)(infinity) alpha(i) epsilon(n-i), where the epsilon(i) are i.i.d. with mean 0 and finite second moment and the ai are either summable or regularly varying with index epsilon (- 1, - 1/2). The sequence {X-n} has short memory in the former case and long memory in the latter. For a large class of functions K, a new approach is proposed to develop both central (root n rate) and noncentral (non-root n rate) limit theorems for S-N = Sigma(n=1)(N) [K(X-n) - EK(X-n)]. Specifically, we show that in the short-memory case the central limit theorem holds for SN and in the long-memory case, S-N can be decomposed into two asymptotically uncorrelated parts that follow a central limit and a noncentral limit theorem, respectively. Further we write the noncentral part as an expansion of uncorrelated components that follow noncentral limit theorems. Connections with the usual Hermite expansion in the Gaussian setting are also explored.