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Asymptotic analysis of average case approximation complexity of Hilbert space valued random elements. (English) Zbl 1333.60108
Summary: We study approximation properties of sequences of centered random elements \(X_d\), \(d \in \mathbb{N}\), 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 \(n^{X_d}(\varepsilon)\) is defined as the minimal number of continuous linear functionals that is needed to approximate \(X_d\) with a relative 2-average error not exceeding a given threshold \(\varepsilon \in(0, 1)\). In the paper we investigate \(n^{X_d}(\varepsilon)\) for arbitrary fixed \(\varepsilon \in (0,1)\) and \(d \to \infty\). Namely, we find criteria for the (un)boundedness of \(n^{X_d}(\varepsilon)\) on \(d\) and for tending \(n^{X_d}(\varepsilon) \to \infty\), \(d \to \infty\), for any fixed \(\varepsilon \in(0, 1)\). In the latter case we obtain necessary and sufficient conditions for the following logarithmic asymptotics \[ \ln n^{X_d}(\varepsilon) = a_d + q(\varepsilon) b_d + o(b_d), \;\;d \to \infty \] at continuity points of a non-decreasing function \(q \colon (0,1) \to \mathbb{R}\). Here \((a_d)_{d \in \mathbb{N}}\) is a sequence and \((b_d)_{d \in \mathbb{N}}\) is a positive sequence such that \(b_d \to \infty\), \(d \to \infty\). 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 the covariance operator.

MSC:
60G99 Stochastic processes
60G60 Random fields
60G15 Gaussian processes
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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