Large Deviations for Quadratic Functionals of Gaussian Processes

Abstract

The Large Deviation Principle (LDP) is derived for several quadratic additive functionals of centered stationary Gaussian processes. For example, the rate function corresponding to \(1/T\int {_0^T } X_t^2 dt\) is the Fenchel-Legendre transform of \(L(y) = - (1/4\pi )\int {_{ - \infty }^\infty } \log (1 - 4\pi yf(s))ds\) where Xt is a continuous time process with the bounded spectral density f(s). This spectral density condition is strictly weaker than the one necessary for the LDP to hold for all bounded continuous functionals. Similar results are obtained for the energy of multivariate discrete-time Gaussian processes and in the regime of moderate deviations, the latter yielding the corresponding Central Limit Theorems.