## The Annals of Probability

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

Emmanuel Rio

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 23, Number 3 (1995), 1188-1203.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176988179

Digital Object Identifier
doi:10.1214/aop/1176988179

Mathematical Reviews number (MathSciNet)
MR1349167

Zentralblatt MATH identifier
0833.60024

JSTOR