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The functional law of the iterated logarithm for stationary strongly mixing sequences. (English) Zbl 0833.60024

Summary: 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 \(\alpha(t) = \alpha_{[t]}\) and we denote by \(Q\) the quantile function of \(|X_0|\). Assume that \[ \int^1_0 \alpha^{-1} (t) Q^2 (t) dt < \infty,\tag{*} \] 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 (*). Moreover, it follows from P. Doukhan, P. Massart and the author [Ann. Inst. Henri Poincaré, Probab. Stat. 30, No. 1, 63-82 (1994; Zbl 0790.60037)] 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 (*) 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.

MSC:

60F05 Central limit and other weak theorems
60F17 Functional limit theorems; invariance principles
60F15 Strong limit theorems

Citations:

Zbl 0790.60037
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