Asymptotic analysis of average case approximation complexity of Hilbert space valued random elements

Abstract

We study approximation properties of sequences of centered random elements Xd, d∈N, with values in separable Hilbert spaces. We focus on sequences of tensor product-type and, in particular, degree-type random elements, which have covariance operators of a corresponding tensor form. The average case approximation complexity nXd(ε) is defined as the minimal number of continuous linear functionals that is needed to approximate Xd with a relative 2-average error not exceeding a given threshold ε∈(0,1). In the paper we investigate nXd(ε) for arbitrary fixed ε∈(0,1) and d→∞. Namely, we find criteria of (un) boundedness for nXd(ε) on d and of tending nXd(ε)→∞, d→∞, for any fixed ε∈(0,1). In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptoticslnnXd(ε)=ad+q(ε)bd+o(bd),d→∞, at continuity points of a non-decreasing function q:(0,1)→R. Here (ad)d∈N is a sequence and (bd)d∈N is a positive sequence such that bd→∞, d→∞. Under rather weak assumptions, we show that for tensor product-type random elements only special quantiles of self-decomposable or, in particular, stable (for tensor degrees) probability distributions appear as functions q in the asymptotics.We apply our results to the tensor products of the Euler integrated processes with a given variation of smoothness parameters and to the tensor degrees of random elements with regularly varying eigenvalues of covariance operator.