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April, 1995 A Maximal Inequality and Dependent Marcinkiewicz-Zygmund Strong Laws
Emmanuel Rio
Ann. Probab. 23(2): 918-937 (April, 1995). DOI: 10.1214/aop/1176988295


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 $(X_i)_{i\in\mathbb{Z}$ with sequence of mixing coefficients $(\alpha_n)_{n\geq 0}$, the Marcinkiewicz-Zygmund SLLN of order $p$ holds if $\int^1_0\lbrack\alpha^{-1}(t)\rbrack^{p-1}Q^p(t)dt < \infty,$ where $\alpha^{-1}$ denotes the inverse function of the mixing rate function $t \rightarrow \alpha_{\lbrack t\rbrack}$ 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.


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Emmanuel Rio. "A Maximal Inequality and Dependent Marcinkiewicz-Zygmund Strong Laws." Ann. Probab. 23 (2) 918 - 937, April, 1995.


Published: April, 1995
First available in Project Euclid: 19 April 2007

zbMATH: 0836.60026
MathSciNet: MR1334177
Digital Object Identifier: 10.1214/aop/1176988295

Primary: 60F15
Secondary: 60E15

Keywords: $\beta$-mixing sequences , Kolmogorov's maximal inequality , Marcinkiewicz-Zygmund strong law of large numbers , Stationary sequences , Strong law of large numbers , strongly mixing sequences , Weak dependence

Rights: Copyright © 1995 Institute of Mathematical Statistics

Vol.23 • No. 2 • April, 1995
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