### Abstract

A large-deviation principle (LDP) at level 1 for random means of the type

is established. The random process {Zn}n≥0 is given by Zn = Φ(Xn) + ξn, n = 0, 1, 2,…, where {Xn}n≥0 and {ξn}n≥0 are independent random sequences: the former is a stationary process defined by Xn = Tn(X0), X0 is uniformly distributed on the circle S1, T: S1 → S1 is a continuous, uniquely ergodic transformation preserving the Lebesgue measure on S1, and {ξn}n≥0 is a random sequence of independent and identically distributed random variables on S1; Φ is a continuous real function. The LDP at level 1 for the means Mn is obtained by using the level 2 LDP for the Markov process {Vn = (Xn, ξn, ξn+1)}n≥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 measure-preserving Lebesgue measure, Φ: [0, 1) → ℛ is a real measurable function, and ξn are independent and identically distributed random variables on ℛ (for instance, they could have a Gaussian distribution with mean zero and variance σ2). The analogous result for the case of autocovariance of order k is also true.