Abstract
A large-deviation principle (LDP) at level 1 for random means of the type
is established. The random process {Z n} n≥0 is given by Z n = Φ(X n) + ξ n , n = 0, 1, 2,..., where {X n} n≥0 and {ξ n} n≥0 are independent 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 {ξn} n≥0 is a random sequence of independent and identically distributed random variables on S 1; Φ is a continuous real function. The LDP at level 1 for the means M n is obtained by using the level 2 LDP for the Markov process {V n = (X n, ξ n , ξ n+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, \(\Phi :\left[ {0,\left. 1 \right)} \right. \to \mathbb{R}\) is a real measurable function, and ξ n are independent and identically distributed random variables on \(\mathbb{R}\) (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.
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Carmona, S.C., Landim, C., Lopes, A. et al. A Level 1 Large-Deviation Principle for the Autocovariances of Uniquely Ergodic Transformations with Additive Noise. Journal of Statistical Physics 91, 395–421 (1998). https://doi.org/10.1023/A:1023008608738
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DOI: https://doi.org/10.1023/A:1023008608738