Asymptotic expansions for sums of block-variables under weak dependence

• 成果类型: Article
• 署名作者: Lahiri, S. N.
• 署名单位: Iowa State University
• 刊物名称: ANNALS OF STATISTICS
• ISSN/ISSBN: 0090-5364
• DOI: 10.1214/009053607000000190
• 发表日期: 2007
• 页码: 1324-1350
• 关键词: central limit-theorem; Edgeworth Expansion; stationary observations; random vectors; bootstrap; CONVERGENCE; statistics; sequence

摘要

Let {X-i}(i)(infinity)=-infinity be a sequence of random vectors and Y-in= f(in)(X-i,X-l) be zero mean block-variables where X-i,X-l = (X-i,X-...,X- Xi+l- 1), i >= 1, are overlapping blocks of length e and where fin are Borel measurable functions. This paper establishes valid joint asymptotic expansions of general orders for the joint distribution of the sums Sigma(n)(i=1) X-i and Sigma(n)(i-1) Y-in under weak dependence conditions on the sequence {X-i}(i=-infinity)(infinity) when the block length l grows to infinity. In contrast to the classical Edgeworth expansion results where the terms in the expansions are given by powers of n(-1/2), the expansions derived here are mixtures of two series, one in powers of n(-1/2) and the other in powers of [n/l](-1/2). Applications of the main results to (i) expansions for 7 Studentized statistics of time series data and (ii) second order correctness of the blocks of blocks bootstrap method are given.