A LEVEL-1 LARGE-DEVIATION PRINCIPLE FOR THE AUTOCOVARIANCES OF UNIQUELY ERGODIC TRANSFORMATIONS WITH ADDITIVE NOISE

A large-deviation principle (LDP) at level 1 for random means of the t ype M-n = 1/n Sigma(f = 0)(n - 1) Z(j)Z(j + 1), n = 1, 2,... is establ ished. The random process {Z(n)}(n greater than or equal to 0) is give n by Z(n) = Phi(X-n) + xi(n), n = 0, 1, 2,..., where {X-n}(n greater t han or equal to 0) and {xi(n)}(n greater than or equal to 0) are indep endent random sequences: the former is a stationary process defined by X-n = T-n(X-0), X-0 is uniformly distributed on the circle S-1, T: S- 1 --> S-1 is a continuous, uniquely ergodic transformation preserving the Lebesgue measure on S-1, and {xi(n)}(n greater than or equal to 0) is a random sequence of independent and identically distributed rando m variables on S-1; Phi is a continuous real function. The LDP at leve l 1 for the means M-n is obtained by using the level 2 LDP for the Mar kov process {V-n = (X-n, xi(n), xi(n + 1))}(n greater than or equal to 0) and the contraction principle. For establishing this level 2 LDP, one can consider a more general setting: T: [0, 1) --> [0, 1) is a mea sure-preserving Lebesgue measure, Phi: [0, 1) --> R is a real measurab le function, and xi(n) are independent and identically distributed ran dom variables on R (for instance, they could have a Gaussian distribut ion with mean zero and Variance sigma(2)). The analogous result for th e case of autocovariance of order ic is also true.