An Estimate of Average Case Approximation Complexity for Tensor Degrees of Random Processes

We consider random fields that are tensor degrees of a random process of second order with a continuous covariance function. The average case approximation complexity of a random field is defined as the minimal number of evaluations of linear functionals needed to approximate the field with a relative twofold average error not exceeding a given threshold. In the present paper, we estimate the growth of average case approximation complexity of random field for an arbitrarily high parametric dimension and for an arbitrarily small error threshold. Using rather weak assumptions concerning the spectrum of covariance operator of the generating random process, we obtain the necessary and sufficient condition that the average case approximation complexity has an upper estimate of special form. We show that this condition covers a wide class of cases and the order of the estimate of the average case approximation complexity coincides with the order of its asymptotic representation obtained by Lifshits and Tulyakova earlier.