Approximation complexity of additive random fields

Let X(t,ω)X(t,ω) be an additive random field for (t,ω)∈[0,1]d×Ω(t,ω)∈[0,1]d×Ω. We investigate the complexity of finite rank approximationX(t,ω)≈∑k=1nξk(ω)ϕk(t).The results are obtained in the asymptotic setting d→∞d→∞ as suggested by Woźniakowski [Tractability and strong tractability of linear multivariate problems, J. Complexity 10 (1994) 96–128.]; [Tractability for multivariate problems for weighted spaces of functions, in: Approximation and Probability. Banach Center Publications, vol. 72, Warsaw, 2006, pp. 407–427.]. They provide quantitative version of the curse of dimensionality: we show that the number of terms in the series needed to obtain a given relative approximation error depends exponentially on d  . More precisely, this dependence is of the form VdVd, and we find the explosion coefficient V.