Decomposition of Additive Random Fields

Abstract

We consider in this work an additive random field on [0, 1]d, which is a sum of d uncorrelated random processes. We assume that the processes have zero mean and the same continuous covariance function. There is a significant interest in the study of random fields of this type. For example, they arise in the theory of intersections and self-intersections of Brownian processes, in the problems concerning small ball probabilities, and in the finite-rank approximation problems with arbitrarily large parametric dimension d. In problems of the last kind, the spectral characteristics of the covariance operator play a key role. For a given additive random field, the dependence of eigenvalues of its covariance operator on eigenvalues of the covariance operator of the marginal processes is quite simple, provided that the identical 1 is an eigenvector of the latter operator. In the opposite case, the dependence is complex, and, therefore, it is hard to study these random fields. Here, summands of the decomposition of the random field into the sum of its integral and its centered version are orthogonal in L2([0, 1]d), but, in general, they are correlated. In the present paper, we propose another interesting decomposition for random fields (it was discovered by the authors while resolving finite-rank approximation problems in the average-case setting). In the obtained decomposition, the summands are orthogonal in L2([0, 1]d) and are uncorrelated. Moreover, for large d, they are respectively close to the integral and to the centered version of the random field with small relative mean square errors.