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A fixed point approach to model random fields. (English) Zbl 1140.60029

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.

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
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