A MAXIMAL INEQUALITY AND DEPENDENT MARCINKIEWICZ-ZYGMUND STRONG LAWS

• 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/1176988295
• 发表日期: 1995
• 页码: 918-937
• 关键词: mixing sequences; CONVERGENCE

摘要

This paper contains some extension of Kolmogorov's maximal inequality to dependent sequences. Next we derive dependent Marcinkiewicz-Zygmund type strong laws of large numbers from this inequality. In particular, for stationary strongly mixing sequences (X(i))(i) (is an element of) (Z) with sequence of mixing coefficients (alpha(n))(n greater than or equal to 0), the Marcinkiewicz-Zygmund SLLN of order p holds if integral(0)(1)[alpha(-1)(t)](p-1)Q(p)(t) dt < infinity, where alpha(-1) denotes the inverse function of the mixing rate function t --> alpha([t]) and Q denotes the quantile function of \X(0)\. The condition is obtained by an interpolation between the condition of Doukhan, Massart and Rio implying the CLT (p = 2) and the integrability of \X(0)\ implying the usual SLLN (p = 1). Moreover, we prove that this condition cannot be improved for stationary sequences and power-type rates of strong mixing.