Variations de champs gaussiens stationnaires: application a l'identification

Abstract

Let $$X_t ,t \in \mathbb{R}^d $$ be a stationary Gaussian random field, with covariance R. For d=1 and d=2, families of variations are described. The convergence in mean square of these variations and a subsequent identification of a model for X are studied. Under suitable glocal conditions for R, the behaviour of these variations depends on the local behaviour of R near the origin. The differences between the case d=1 and d=2 are particularly emphasised: for d=1, there exists only one variation; for d=2, several families of variations are available which provided a useful tool for identifying different models: for example, Orstein-Uhlenbeck processes can be identified in mean square on $$\mathbb{R}$$ , but not on $$\mathbb{R}$$ .