Small values of Gaussian processes and functional laws of the iterated logarithm (Q1346969)

Let \(\{X(t), t \geq 0\}\) be a Gaussian process with stationary increments. Assuming that \(X(t)\) is locally nondeterministic the authors establish Chung's type of the law of the iterated logarithm. To be precise, they show that the limit inferior of \(X(t)\), properly normalized, is almost surely a constant \(c > 0\) as \(t \to 0\) \((t \to \infty)\). Unfortunately the authors have not been able to compute \(c\). The results for fractional Brownian motion follow as special cases. In particular, when \(X(t)\) is a fractional Brownian motion, Strassen's functional law of the iterated logarithm is established, which at the same time gives a rate of convergence. This is an extension of the result for standard Brownian motion by \textit{E. Csáki} [Z. Wahrscheinlichkeitstheorie Verw. Geb. 54, 287-301 (1980; Zbl 0441.60027)].