Large deviations and the rate distortion theorem for Gibbs distributions

Large deviation theory is used to obtain the rate distortion theorem for Gibbs distributions together with exponentially small error probabilities. Large deviation theorems provide asymptotically exponential upper and lower bounds on the probability that the empirical distribution under a Gibbs distribution deviates in variational norm from the marginal. In particular these hold if the Gibbs distribution is a product measure. Using these theorems many of the standard asymptotic results of errorless coding theory can be neatly formulated and extended to Gibbs random fields. We present the application of these theorems to coding with distortion.