A Maximal Inequality and Dependent Marcinkiewicz-Zygmund Strong Laws

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 $(X_i)_{i\in\mathbb{Z}$ 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.