**Optimal low-dimensional approximations of random vector functions
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The paper considers approximations of first- and second-order moments of random functions with values in a high-dimensional Euclidean space using projections onto suitable low-dimensional linear submanifolds. To quantify the goodness of the approximation a criterion based on the mean squared Euclidean distance is introduced. In case of wide-sense stationary random functions optimal low-dimensional linear submanifolds are given in terms of the mean vector and eigenvectors of the variance matrix.