Lim inf results for Gaussian samples and Chung's functional LIL
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.
scientific article; zbMATH DE number 828370