On the maximum of a fractional Brownian motion

Let $b_{\gamma}(t)$, $b_{\gamma}(0)=0$ be a fractional Brownian motion, i.e., a Gaussian process with the structural function $\mathbf{E}|b_{\gamma}(t)-b_{\gamma}(s)|^2=|t-s|^\gamma$, $0 < \gamma < 2$. The logarithmic asymptotics as $T\to\infty$ is found for the probabilities $P_T=\mathsf{P}\{b_{\gamma}(t)<1,\ -\rho T0\}$ this asymptotics is independent of $\gamma$.