On the Large Deviation Principle for Stationary Weakly Dependent Random Fields

Abstract

The large deviation principle for the empirical field of a stationary $\mathbb{Z}^d$-indexed random field is proved under strong mixing dependence assumptions. The strong mixing coefficients considered allow us to separate the ratio-mixing condition used in the literature into a part directly responsible for the (nonuniform) large deviation principle and another one, which is used when the state space is noncompact. Results are applied to obtain variants of recent large deviation theorems for Markov chains and for Gibbs fields. The proofs are based on a new criterion for the large deviation principle which is stated in Appendix C.