We first analyze the simple Kriging task, the flagship problem in Geostatistics, from a statistical learning perspective, i.e. by carrying out a non-parametric finite-sample predictive analysis. In this context, the standard probabilistic theory of statistical learning does not apply directly and theoretical guarantees of the generalization capacity of the Kriging predictive rule learned from spatial data are left to be established. Given a finite number of values taken by a realization of a square integrable random field, with unknown covariance structure, the goal is to predict the unknown values that the random field takes at any other location in the spatial domain with minimum quadratic risk. Establishing the generalization capacity of empirical risk minimizer is far from straightforward, due to the non independent and identically distributed nature of the training data involved in the learning procedure. In the first part of this thesis, non-asymptotic bounds are proved for the excess risk of a plug-in predictive rule mimicking the true minimizer in the case of isotropic stationary Gaussian processes, observed at locations forming a regular grid in the learning stage. These theoretical results, as well as the role played by the technical conditions required to establish them, are illustrated by various numerical experiments, on simulated data and on real-world datasets, and may hopefully pave the way for further developments in statistical learning based on spatial data.

In the second part of this thesis, we focus on space-time Hawkes processes. Many modern spatio-temporal data sets, in sociology, epidemiology or seismology, for example, exhibit self-exciting characteristics, with simultaneous triggering and clustering behaviors, that a suitable spatio-temporal Hawkes process can accurately capture. However, dealing efficiently with the high volumes of data now available is challenging. We aim at developing a fast and flexible parametric inference technique to recover the parameters of the kernel functions involved in the intensity function of a spatio-temporal Hawkes process based on such data. Our statistical approach combines three key ingredients: (1) kernels with finite support are considered, (2) the space-time domain is appropriately discretized, and (3) (approximate) precomputations are used. The inference technique we propose consists of a fast and statistically accurate solver. In addition to describing the algorithmic aspects, numerical experiments have been carried out on synthetic and real spatio-temporal data, providing solid empirical evidence of the relevance of the proposed methodology.