THE FUNCTIONAL LAW OF THE ITERATED LOGARITHM FOR STATIONARY STRONGLY MIXING SEQUENCES

• 成果类型: Article
• 署名作者: RIO, E
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176988179
• 发表日期: 1995
• 页码: 1188-1203
• 关键词: central limit-theorem; random-variables

摘要

Let (X(i))(i is an element of Z) be a strictly stationary and strongly mixing sequence of real-valued mean zero random variables. Let (alpha(n))(n>0) be the sequence of strong mixing coefficients. We define the strong mixing fraction alpha (.) by alpha(t) = alpha([t]) and we denote by Q the quantile function of \X(0)\. Assume that (*) integral(0)(1) alpha(-1)(t)Q(2)(t)dt < infinity, where f(-1) denotes the inverse of the monotonic function f. The main result of this paper is that the functional law of the iterated logarithm (LIL) holds whenever (X(i))(i is an element of z) satisfies (*). Moreover, it follows from Doukhan, Massart and Rio that for any positive a there exists a stationary sequence (X(i))(i is an element of z) with strong mixing coefficients alpha(n) of the order of n(-a) such that the bounded LIL does not hold if condition (*) is violated. The proof of the functional LIL is mainly based on new maximal exponential inequalities for strongly mixing processes, which are of independent interest.