Asymptotic Inference for Markov Random Fields on ℤd

The purpose of this paper is to generalize results of D.K. PICKARD [1,2,3] on asymptotic inference for the Ising model. Using properties of Gibbs measures on ℤd, we give the canonical exponential structure in which we can study the estimation problem, in the case of vertical sampling. This exponential structure depends on the range of the interaction potential of the underlying Markov random field; we construct, therefore, a test to measure this range: this test is based on the markovian character of the associated Gibbs measure; next we present the properties and the asymptotic laws of the estimators and we compare horizontal to vertical sampling [4]. Certain properties depend on the criticality of the interaction potential [5,6]: consequently, we construct a test of criticality for the interaction potential. Finally, we set the problem of estimation in a random Ising model.