Sample Path Large Deviations for Squares of Stationary Gaussian Processes

Abstract : In this paper, we show large deviations for random step functions of type Zn(t) = 1 n X[nt] k=1 X2 k ; where fXkgk is a stationary Gaussian process. We deal with the associated random measures n = 1 n Pn k=1 X2 k k=n. The proofs require a Szego theorem for generalized Toeplitz matrices, which is presented in the Appendix and is analogous to a result of Kac, Murdoch and Szego [10]. We also study the polygonal line built on Zn(t) and show moderate deviations for both random families.