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Approximation complexity of tensor product-type random fields with heavy spectrum

Authors:	A A Khartov

Institution:	1. St. Petersburg State University, St. Petersburg, Russia

Abstract:	We consider a sequence of Gaussian tensor product-type random fields _d ⁽ⁿ⁾ corresponding to the n maximal eigenvalues λ _k, $n_d^{pr} (\varepsilon ,\delta ): = \min \left\{ {n \in \mathbb{N}:\mathbb{P}(\left\\| {X_d - X_d^{(n)} } \right\\|_{2,d}^2 > \varepsilon ^2 \mathbb{E}\left\\| {X_d } \right\\|_{2,d}^2 ) \leqslant \delta } \right\},$ and the probabilistic approximation complexity $n_d^{avg} (\varepsilon ): = \min \left\{ {n \in \mathbb{N}:\mathbb{E}\left\\| {X_d - X_d^{(n)} } \right\\|_{2,d}^2 \leqslant \varepsilon ^2 \mathbb{E}\left\\| {X_d } \right\\|_{2,d}^2 } \right\}$ , as the parametric dimension d → ∞ the error threshold ? ∈ (0, 1) is fixed, and the confidence level δ = δ(d, ?) is allowed to approach zero. Supplementing recent results of M.A. Lifshits and E.V. Tulyakova, we consider the case where the sequence
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