We prove a central limit theorem for strictly stationary random fields under a projective assumption. Our criterion is similar to projective criteria for stationary sequences derived from Gordin's theorem about approximating martingales. However our approach is completely different, for we establish our result by adapting Lindeberg's method. The criterion that it provides is weaker than martingale-type conditions, and moreover we obtain as a straightforward consequence, central limit theorems for α-mixing or φ-mixing random fields.

A central limit theorem for stationary random fields

Large deviations for martingales

We establish the asymptotic normality of the regression estimator in a fixed-design setting when the errors are given by a field of dependent random variables. The result applies to martingale-difference or strongly mixing random fields. On this basis, a statistical test that can be applied to image analysis is also presented.

https://hal.archives-ouvertes.fr/hal-00403055/document

Asymptotic normality of kernel estimates in a regression model for random fields

Invariance principle for martingale-difference random fields

We prove a central limit theorem for conditionally centred random fields, under a moment condition and strict positivity of the empirical variance per observation. We use a random normalization, which fits non-stationary situations. The theorem applies directly to Markov random fields, including the cases of phase transition and lack of stationarity. One consequence is the asymptotic normality of the maximum pseudo-likelihood estimator for Markov fields in complete generality.

A central limit theorem for conditionally centred random fields with an application to Markov fields

/pdf/martingale-difference-gibbs-random-fields-and-central-limit-4zezh8d6m4.pdf

A.N. Petrosian

Papers

Martingale-difference Gibbs random fields and central limit theorem

Migration and Housing Construction in the Regional Capitals of Russia and Their Suburbs