Sample Path Large Deviations for Squares of Stationary Gaussian Processes

Abstract

In this paper, we show large deviations for random step functions of type $Z_n(t)=\frac{1}{n}\sum_{k=1}^{[nt]}X_k^2,$ where $\{X_k\}_k$ is a stationary Gaussian process. We deal with the associated random measures $\nu_n=\frac{1}{n}\sum_{k=1}^nX_k^2 \delta_{k/n}$. The proofs require a Szegö theorem for generalized Toeplitz matrices which is analogous to a result of Kac, Murdoch, and Szegö [J. Rational Mech. Anal., 2 (1953), pp. 767--800]. We also study the polygonal line built on $Z_n(t)$ and show moderate deviations for both random families.