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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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