## The Annals of Probability

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

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

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60F05: Central limit and other weak theorems

**Keywords**

Functional law of the iterated logarithm maximal exponential inequalities moment inequalities strongly mixing sequences strong invariance principle stationary sequences

#### Citation

Rio, Emmanuel. The Functional Law of the Iterated Logarithm for Stationary Strongly Mixing Sequences. Ann. Probab. 23 (1995), no. 3, 1188--1203. doi:10.1214/aop/1176988179. https://projecteuclid.org/euclid.aop/1176988179