Asymptotic Properties of the R/S Statistics for Linear Processes

Abstract

Let {X _k ; k ≥ 1} be a linear process defined by where {ϵ _i ; − ∞ <i < ∞} is a doubly infinite sequence of i.i.d. random variables, and . Denote the R/S statistic by Q(n) = R(n)/S(n), where , and are the adjust range of partial sums, the sample mean, and the sample variance, respectively. In this article, the exact moment convergence rates in the law of the iterated logarithm, the law of the logarithm and the complete convergence for R/S statistics are achieved, when may be infinite.

Keywords:

• Brownian bridge
• Complete convergence
• Law of the iterated logarithm
• Linear process
• R/S statistics

Mathematics Subject Classification:

• (PR) 60F15
• 60G50

Acknowledgment

Project supported by National Natural Science Foundation of China (Nos. 10901138 and 11026087), Natural Science Foundation of Zhejiang Province (No. Y6100663) and the Introduction Talent Foundation of Zhejiang Gongshang University (1020XJ200961).