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/n Sigma([nt])(k=1) X-k(2), where {X-k}(k) is a stationary Gaussian process. We deal with the associated random measures nu(n) = 1/n Sigma(n)(k=1) X-k(2)delta(k/n). The proofs require a Szego theorem for generalized Toeplitz matrices which is analogous to a result of Kac, Murdoch, and Szego [J. Rational Mech. Anal., 2 (1953), pp. 767-800]. We also study the polygonal line built on Zn(t) and show moderate deviations for both random families.