Asymptotic analysis in multivariate average case approximation with Gaussian kernels

We consider tensor product random fields Yd, d ∈ N, whose covariance funtions are Gaussian kernels. The average case approximation complexity nYd(ε) is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate Yd, with relative 2-average error not exceeding a given threshold ε ∈ (0, 1). We investigate the growth of nYd(ε) for arbitrary fixed ε ∈ (0, 1) and d → ∞. Namely, we find criteria of boundedness for nYd(ε) on d and of tending nYd(ε) → ∞, d → ∞, for any fixed ε ∈ (0, 1). In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptotics lnnd(ε) = ad + q(ε)bd + o(bd), d → ∞, with any ε ∈ (0, 1). Here q : (0, 1) → R is a non-decreasing function, (ad)d∈N is a sequence and (bd)d∈N is a positive sequence such that bd → ∞, d → ∞. We show that only special quantiles of self-decomposable distribution functions appear as functions q in a given asymptotics.