Law of the iterated logarithm for stationary processes

• 成果类型: Article
• 署名作者: Zhao, Ou; Woodroofe, Michael
• 署名单位: University of Michigan System; University of Michigan
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117907000000079
• 发表日期: 2008
• 页码: 127-142
• 关键词: CENTRAL-LIMIT-THEOREM; additive-functionals; mixing sequences; random-variables

摘要

There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes ...., X-1, X-0, X-1 ,... whose partial sums S-n = X-1 + ... + X-n are of the form S-n = M-n + R-n, where M-n is a square integrable martingale with stationary increments and R-n is a remainder term for which E(R-n(2)) = o(n). Here we explore the law of the iterated logarithm (LIL) for the same class of processes. Letting parallel to center dot parallel to denote the norm in L-2(P), a sufficient condition for the partial sums of a stationary process to have the form S-n = M-n + R-n is that n(-3/2)parallel to E(S-n vertical bar X-0, X-1....)parallel to be summable. A sufficient condition for the LIL is only slightly stronger, requiring n(-3/2) log(3/2)(n)parallel to E(S-n vertical bar X-0, X-1....)parallel to to be summable. As a by-product of our main result, we obtain an improved statement of the conditional central limit theorem. Invariance principles are obtained as well.