Article révisé par les pairs
Résumé : This paper develops a framework for the estimation of the functional mean and the functional principal components when the functions form a random field. More specifically, the data we study consist of curves $X(s_k;t)$, $t ∈ [0, T]$, observed at spatial points $s_1, s_2, ldots , s_N$. We establish conditions for the sample average (in space) of the $X(s_k)$ to be a consistent estimator of the population mean function, and for the usual empirical covariance operator to be a consistent estimator of the population covariance operator. These conditions involve an interplay of the assumptions on an appropriately defined dependence between the functions $X(s_k)$ and the assumptions on the spatial distribution of the points sk. The rates of convergence may be the same as for iid functional samples, but generally depend on the strength of dependence and appropriately quantified distances between the points sk. We also formulate conditions for the lack of consistency