Doukhan, Paul; Truquet, Lionel A fixed point approach to model random fields. (English) Zbl 1140.60029 ALEA, Lat. Am. J. Probab. Math. Stat. 3, 111-132 (2007). The authors prove the existence and uniqueness of the solution of the equation \[ X_{t}=F((X_{t-j})_{j\in\mathbb{Z}^{d}\setminus\{0\}};\xi_{t}), \] where the input \((\xi_{t})\) is an i.i.d. random field. The proofs rely on the contraction principle hence Lipschitz type conditions are needed, but there are no assumptions relative to the conditional distribution. The solution writes \(X_{t}=H((\xi_{t-s})_{s\in\mathbb{Z}^{d}}\) and is a stationary random field with infinite interactions. The authors prove weak dependence properties of this solution. General results for stationary (non necessarily independent) input are also stated. Those results imply heavy restrictions on the innovations in some cases: a convenient notion of causality is thus used. Reviewer: Mihai Gradinaru (Rennes) Cited in 11 Documents MSC: 60G60 Random fields 60B12 Limit theorems for vector-valued random variables (infinite-dimensional case) 60F25 \(L^p\)-limit theorems 60K35 Interacting random processes; statistical mechanics type models; percolation theory 62M40 Random fields; image analysis 60B99 Probability theory on algebraic and topological structures 60K99 Special processes Keywords:random fields; limit theorems for vector-valued random variables; interacting random processes; weak dependence; Bernoulli shifts PDFBibTeX XMLCite \textit{P. Doukhan} and \textit{L. Truquet}, ALEA, Lat. Am. J. Probab. Math. Stat. 3, 111--132 (2007; Zbl 1140.60029)