Consistency of the mean and the principal components of spatially distributed functional data

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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(sₖ;t),t∈[0,T], observed at spatial points s₁,s₂,,sₙ. We establish conditions for the sample average (in space) of the X(sₖ) 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(sₖ) and the assumptions on the spatial distribution of the points sₖ. The rates of convergence may be the same as for i.i.d. 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.