Some liminf results on increments of fractional Brownian motion
Let \(\{X(t), t \geq 0\}\) be a fractional Brownian motion of order \(2 \alpha\) with \(0 < \alpha < 1\), i.e. \(\{X(t), t \geq 0\}\) is a real-valued Gaussian process with mean zero, \(X(0) = 0\) and \(EX^2 (t) = t^{2 \alpha}\) \((t \geq 0)\). Define \(I_1 (T) = \sup_{0 \leq t \leq T - a_T} |X (t + a_T) - X(t) |\) and \(I_2 (T) = \sup_{0 \leq t \leq T - a_T} \sup_{0 \leq s \leq a_T} |X(t + s) - X(t) |\), where \(a_T\) \((0 < a_T \leq T)\) is a function of \(T\). The work shows that if \(a_T\) satisfies (i) \(a_T\) and \(T/a_T\) are nondecreasing and (ii) \(\lim_{T \to \infty} T/(a_T \log \log T) = \infty\), then \[ \liminf_{T \to \infty} \gamma (T) I_1 (T) = \liminf_{T \to \infty} \gamma (T) I_2 (T) = 1 \text{ a.s.} \qquad (0 < \alpha \leq 1/2) \] and \[ \sqrt {1 - 4^{\alpha - 1}} \leq \liminf_{T \to \infty} \gamma (T) I_1 (T) \leq \liminf_{T \to \infty} \gamma (T) I_2 (T) \leq 1 \text{ a.s.} \qquad (1/2 < \alpha < 1), \] where \(\gamma (T) = a_T^{- \alpha} \{2 (\log (T/a_T) - \log \log \log T) \}^{- 1/2}\); whereas if the condition (ii) is replaced by \(\lim_{T \to \infty} T/(a_T \log \log T) = 0\), then \[ c_1 \leq \liminf_{T \to \infty} \gamma' (T) I_2 (T) \leq c_2 \text{ a.s.}, \] where \(\gamma' (T) = \{a_T \log (1 + T/(a_T \log \log T)) \}^{- \alpha}\) and \(c_1\), \(c_2\) are two positive constants depending only on \(\alpha\). These results extend the previous theorems by \textit{E. Csáki} and \textit{P. Révész} [Acta Math. Acad. Sci. Hungar. 33, 37-49 (1979; Zbl 0392.60056)], \textit{Q. Shao} [J. Math., Wuhan Univ. 6, 175-182 (1986; Zbl 0616.60035)] and others. Later, the author [Acta Math. Hung. 76 (to appear)] showed that \[ \liminf_{T \to \infty} \gamma (T) I_1 (T) = \liminf_{T \to \infty} \gamma (T) I_2 (T) = 1 \text{ a.s.} \] remains true for all \(0 < \alpha < 1\) whenever the condition (ii) is satisfied.
scientific article; zbMATH DE number 897260