Spatial analysis methods have seen a rapid rise in popularity due to demand from a wide range of fields. These include, among others, biology, spatial economics, image processing, environmental and earth science, ecology, geography, epidemiology, agronomy, forestry and mineral prospection.In spatial problems, observations come from a spatial process X = {X s , s ∈ S} indexed by a spatial set S, with X s taking values in a state space E. The positions of observation sites s ∈ S are either fixed in advance or random. Classically, S is a 2-dimensional subset, S ⊆ R 2 . However, it could also be 1-dimensional (chromatography, crop trials along rows) or a subset of R 3 (mineral prospection, earth science, 3D imaging). Other fields such as Bayesian statistics and simulation may even require spaces S of dimension d ≥ 3. The study of spatial dynamics adds a temporal dimension, for example (s,t) ∈ R 2 ×R + in the 2-dimensional case. This multitude of situations and applications makes for a very rich subject. To illustrate, let us give a few examples of the three types of spatial data that will be studied in the book.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY We study asymptotic properties of various estimation procedures for a general stationary regular process on a d-dimensional lattice. Differences between d = 1, time series, and d > 2, spatial processes, are pointed out. We suppose that the process is observed on a set PN, with CN points, which tends to infinity with the same speed in all directions. The relative edge effect is of order CN lId, increasing with d: this effect is without statistical importance if d = 1, but is important if d > 2. We give a lCN-consistent, asymptotically normal estimator of the underlying parameter, the procedure being constructed by a modification of Whittle's approximation to the log likelihood. In the Gaussian case, this procedure is asymptotically efficient.

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For a general class of order selection criteria, we establish analytic and nonasymptotic evaluations of both the underfitting and overfitting sets of selected models. These evaluations are further specified in various situations including regressions and autoregressions with finite or infinite variances. We also show how upper bounds for the misfitting probabilities and hence conditions ensuring the weak consistency can be derived from the given evaluations. Moreover, it is demonstrated how these evaluations, combined with a law of the iterated logarithm for some relevant statistic, can provide conditions ensuring the strong consistency of the model selection criterion used. Academic PressAMS 1991 subject classifications: 62F12, 62M10, 62M40.

Let Y be an Ornstein-Uhlenbeck diffusion governed by a stationary and ergodic process X. We give condition for ergodicity of Y and give conditions that ensures existence of moment for the invariant law of Y