The Functional Law of the Iterated Logarithm for Stationary Strongly Mixing Sequences

Rio, Emmanuel

Ann. Probab., Tome 23 (1995) no. 3, p. 1188-1203 / Harvested from Project Euclid

Résumé

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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Publié le : 1995-07-14
Classification:  Functional law of the iterated logarithm,  maximal exponential inequalities,  moment inequalities,  strongly mixing sequences,  strong invariance principle,  stationary sequences,  60F05