  In the Big Data era, with the ubiquity of geolocation sensors in particular,
massive datasets exhibiting a possibly complex spatial dependence structure are
becoming increasingly available. In this context, the standard probabilistic
theory of statistical learning does not apply directly and guarantees of the
generalization capacity of predictive rules learned from such data are left to
establish. We analyze here the simple Kriging task from a statistical learning
perspective, i.e. by carrying out a nonparametric finite-sample predictive
analysis. Given $d\geq 1$ values taken by a realization of a square integrable
random field $X=\{X_s\}_{s\in S}$, $S\subset \mathbb{R}^2$, with unknown
covariance structure, at sites $s_1,\; \ldots,\; s_d$ in $S$, the goal is to
predict the unknown values it takes at any other location $s\in S$ with minimum
quadratic risk. The prediction rule being derived from a training spatial
dataset: a single realization $X'$ of $X$, independent from those to be
predicted, observed at $n\geq 1$ locations $\sigma_1,\; \ldots,\; \sigma_n$ in
$S$. Despite the connection of this minimization problem with kernel ridge
regression, establishing the generalization capacity of empirical risk
minimizers is far from straightforward, due to the non independent and
identically distributed nature of the training data $X'_{\sigma_1},\; \ldots,\;
X'_{\sigma_n}$ involved in the learning procedure. In this article,
non-asymptotic bounds of order $O_{\mathbb{P}}(1/\sqrt{n})$ 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 are illustrated
by various numerical experiments, on simulated data and on real-world datasets.
