Maximum of a Fractional Brownian Motion: Probabilities of Small Values

Let b _γ (t), b _γ(0)= 0 be a fractional Brownian motion, i.e., a Gaussian process with the structure function , 0 < γ < 2. We study the logarithmic asymptotics of P _T = P{b _γ (t) < 1,□t∈TΔ} as T→∞, where Δ is either the interval (0,1) or a bounded region that contains a vicinity of 0 for the case of multidimensional time. It is shown that ln P _T = - D ln T(1 + o(1)), where D is the dimension of zeroes of b _γ (t) in the former case and the dimension of time in the latter. Received: 28 September 1998 / Accepted: 19 February 1999