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July, 1995 The Functional Law of the Iterated Logarithm for Stationary Strongly Mixing Sequences
Emmanuel Rio
Ann. Probab. 23(3): 1188-1203 (July, 1995). DOI: 10.1214/aop/1176988179


Let $(X_i)_{i\in\mathbb{Z}}$ 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 function $\alpha(\cdot)$ by $\alpha(t) = \alpha_{\lbrack t\rbrack}$ and we denote by $Q$ the quantile function of $|X_0|$. Assume that \begin{equation*}\tag{*}\int^1_0\alpha^{-1}(t)Q^2(t) dt < \infty,\end{equation*} 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 $(X_i)_{i\in\mathbb{Z}}$ satisfies $(\ast)$. Moreover, it follows from Doukhan, Massart and Rio that for any positive $a$ there exists a stationary sequence $(X_i)_{i\in\mathbb{Z}}$ with strong mixing coefficients $\alpha_n$ of the order of $n^{-a}$ such that the bounded LIL does not hold if condition $(\ast)$ 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.


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Emmanuel Rio. "The Functional Law of the Iterated Logarithm for Stationary Strongly Mixing Sequences." Ann. Probab. 23 (3) 1188 - 1203, July, 1995.


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

zbMATH: 0833.60024
MathSciNet: MR1349167
Digital Object Identifier: 10.1214/aop/1176988179

Primary: 60F05

Keywords: Functional law of the iterated logarithm , maximal exponential inequalities , Moment inequalities , Stationary sequences , Strong invariance principle , strongly mixing sequences

Rights: Copyright © 1995 Institute of Mathematical Statistics


Vol.23 • No. 3 • July, 1995
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