Rates of Poisson Approximation to Finite Range Random Fields

The Stein-Chen approach is used to obtain bounds on the Poisson approximation of a random field, in both a random variable and a stochastic process sense. The hypotheses are Dobrushin's condition or, alternatively, positive dependence combined with a bound on decay of correlations. Rates of convergence are derived which supplement the limit theorems of Berman. The results have application to certain Gibbs states at both high and low temperature.