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

Emmanuel Rio
The Annals of Probability
Vol. 23, No. 3 (Jul., 1995), pp. 1188-1203
Stable URL: http://www.jstor.org/stable/2244868
Page Count: 16
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The Functional Law of the Iterated Logarithm for Stationary Strongly Mixing Sequences
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Abstract

Let (Xi)i∈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 α(·) by α(t) = α[ t] and we denote by Q the quantile function of |X0|. 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 (Xi)i∈Z satisfies (*). Moreover, it follows from Doukhan, Massart and Rio that for any positive a there exists a stationary sequence (Xi)i∈Z with strong mixing coefficients αn of the order of n-a such that the bounded LIL does not hold if condition (*) 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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