Summary
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 \(\mathbb{R}\).
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Guyon, X. Variations de champs gaussiens stationnaires: application a l'identification. Probab. Th. Rel. Fields 75, 179–193 (1987). https://doi.org/10.1007/BF00354032
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DOI: https://doi.org/10.1007/BF00354032
Keywords
- Covariance
- Stochastic Process
- Probability Theory
- Statistical Theory
- Random Field