On a decomposition of additive random fields

Abstract

We consider 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. They appear for example in the theory of intersections and selfintersections of Brownian processes, in the problems concerning the small ball probabilities, and in the finite rank approximation problems with arbitrary large parametric dimension d. In the last problems the spectral characteristics of the covariance operator play key role. For a given additive random field the eigenvalues of its covariance operator easily depend on the eigenvalues of the covariance operator of the marginal processes in the case, when the latter has identical 1 as an eigenvector. In the opposite case the dependence is complex, that makes these random fields difficult to study. Here decomposing the random field into the sum of its integral and its centered version, the summands will be orthogonal in L2([0, 1]d), but in the general case they are correlated. In the present paper we propose another interesting decomposition for the random field, that was observed by the authors within finite rank approximation problems in the average case setting. In the derived decomposition the summands are orthogonal in L2([0, 1]d) and 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 squared error.