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