A functional LIL and some weighted occupation measure results for fractional Brownian motion
\textit{X. Chen} and the authors [Ann. Probab. 28, No. 1, 258-276 (2000)] proved a functional law of the iterated logarithm for symmetric stable processes with stationary independent increments. Corresponding to the above result, the authors establish a functional law of the iterated logarithm for \(\gamma\)-fractional Brownian motion \(\{B_\gamma (t)\}\) defined by the form \[ B_\gamma(t)= \int_{\mathbb{R}^1} k_\gamma(t,x)dB (x)\quad (t\geq 2). \] Here, \(\{B(t): -\infty <t< \infty\}\) is a sample continuous Brownian motion with \(B(0)=0\) and \(k_\gamma (t, x)\) is some function of \((t,x)\) which depends on \(0<\gamma<2\); specifically, \(k_1(t,x)= I_{[0,1]}(x)\). As consequences, the weighted occupation measure results for the fractional Brownian motion are obtained.
functional law of the iterated logarithm
scientific article; zbMATH DE number 1892256