Large Deviations for Expanding Transformations with Additive White Noise

Abstract

Large-deviations estimates for the autocorrelations of order kof the random process Z _n=φ(X _n)+ξ _n, n≥0, are obtained. The processes (X _n) _n≥0and (ξ _n) _n≥0are independent, ξ _n, n≥0, are i.i.d. bounded random variables, X _n=T ⁿ(X ₀), n∈$\mathbb{N}$, T: M→Mis expanding leaving invariant a Gibbs measure on a compact set M, and φ: M→$\mathbb{R}$is a continuous function. A possible application of this result is the case where Mis the unit circle and the Gibbs measure is the one absolutely continuous with respect to the Lebesgue measure on the circle. The case when Tis a uniquely ergodic map was studied in Carmona et al.(1998). In the present paper Tis an expanding map. However, it is possible to derive large-deviations properties for the autocorrelations samples (1/n) ∑ⁿ⁻¹ _j=0 Z _j Z _j+k . But the deviation function is quite different from the uniquely ergodic case because it is necessary to take into account the entropy of invariant measures for Tas an important information. The method employed here is a combination of the variational principle of the thermodynamic formalism with Donsker and Varadhan's large-deviations approach.