Exact Asymptotics of Large Deviations of Stationary Ornstein—Uhlenbeck Processes for L~p-Functionals, p > 0

摘要

We prove a general result on the exact asymptotics of the probability P{∫ from 0 to 1 of |η_γ(t)|~p dt > u~p} as u → ∞, where p > 0, for a stationary Ornstein-Uhlenbeck process η_γ(t), i.e., a Gaussian Markov process with zero mean and with the covariance function Eη_γ(t)η_γ(s) = e~(-γ|t-s|), t, s ∈ R, γ > 0. We use the Laplace method for Gaussian measures in Banach spaces. Evaluation of constants is reduced to solving an extreme value problem for the rate function and studying the spectrum of a second-order differential operator of the Sturm-Liouville type. For p = 1 and p = 2, explicit formulas for the asymptotics are given.