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Lim inf results for Gaussian samples and Chung’s functional LIL. (English) Zbl 0849.60022

Let \(X, X_1, X_2, \dots\) be i.i.d. centered Gaussian random elements of a Banach space \(B\) with common law \(\mu\) and let \(\psi (\varepsilon) = \log \mu (x : |x |\leq \varepsilon)\). The authors study \[ \liminf_{n \to \infty} d(n)^{-1} |X_n/(2 \log n)^{1/2} - f | \] for elements \(f\) of \(B\) where \(d = d(n)\) is a solution of certain transcendental equations involving \(\psi (d(2 \log n)^{1/2})\) and in addition \(f\) when \(|f |_\mu = 1\); the results differ according to whether \(|f |_\mu < 1\) or \(|f |_\mu = 1\). Conditions are also given for \(d_1 (n) \approx d_2 (n)\) when \(|f_1 |_\mu = |f_2 |_\mu = 1\), and the exact constants are obtained for some examples in Hilbert space.
Reviewer: S.Asmussen (Lund)

MSC:

60F15 Strong limit theorems
60G15 Gaussian processes
60F17 Functional limit theorems; invariance principles
60G17 Sample path properties
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