Asymptotic analysis in multivariate average case approximation with Gaussian kernels

We consider tensor product random fields Yd, d∈N, whose covariance functions are Gaussian kernels with a given sequence of length scale parameters. investigate the growth average case approximation complexity nYd(ε) Yd for arbitrary fixed ε∈(0,1) and d→∞. Namely, we find criteria boundedness depending on d nYd(ε)→∞ when d→∞ any ε∈(0,1). In latter obtain necessary sufficient conditions following logarithmic asymptoticsln⁡nYd(ε)=ad+q(ε)bd+o(bd),d→∞, Here q:(0,1)→R is non-decreasing function, (ad)d∈N (bd)d∈N positive such that bd→∞, show only special quantiles self-decomposable distribution appear as q in asymptotics. These general results apply to under particular assumptions