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A Maximal Inequality and Dependent Marcinkiewicz-Zygmund Strong Laws

Emmanuel Rio
The Annals of Probability
Vol. 23, No. 2 (Apr., 1995), pp. 918-937
Stable URL: http://www.jstor.org/stable/2245012
Page Count: 20
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A Maximal Inequality and Dependent Marcinkiewicz-Zygmund Strong Laws
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Abstract

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 (Xi)_{i∈{Z} with sequence of mixing coefficients (αn)n≥ 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 α-1 denotes the inverse function of the mixing rate function t → α[ t] and Q denotes the quantile function of |X0|. The condition is obtained by an interpolation between the condition of Doukhan, Massart and Rio implying the CLT (p = 2) and the integrability of |X0| 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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