SAMPLE PATH LARGE DEVIATIONS FOR SQUARES OF STATIONARY GAUSSIAN PROCESSES.

In this paper, we show large deviations for random step functions of type Z_n(t) = 1/nΣ[^nt] _k=1 X² _k, where {X_k}_k is a stationary Gaussian process. We deal with the associated random measures vn = 1/n Σⁿ _k=1 X² _kб_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. [ABSTRACT FROM AUTHOR]