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



We consider a sequence of Gaussian tensor product-type random fields
, where Open image in new window and Open image in new window are all positive eigenvalues and eigenfunctions of the covariance operator of the process X1, Open image in new window are standard Gaussian random variables, and Open image in new window is a subset of positive integers. For each d ∈ ℕ, the sample paths of Xd almost surely belong to L2([0, 1]d) with norm ∥·∥2,d. The tuples Open image in new window, are the eigenpairs of the covariance operator of Xd. We approximate the random fields Xd, dOpen image in new window, by the finite sums Xd(n) corresponding to the n maximal eigenvalues λk, Open image in new window.
We investigate the logarithmic asymptotics of the average approximation complexity
$$n_d^{pr} (\varepsilon ,\delta ): = \min \left\{ {n \in \mathbb{N}:\mathbb{P}(\left\| {X_d - X_d^{(n)} } \right\|_{2,d}^2 > \varepsilon ^2 \mathbb{E}\left\| {X_d } \right\|_{2,d}^2 ) \leqslant \delta } \right\},$$
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 ɛ ∈ (0, 1) is fixed, and the confidence level δ = δ(d, ɛ) is allowed to approach zero. Supplementing recent results of M.A. Lifshits and E.V. Tulyakova, we consider the case where the sequence Open image in new window decreases regularly and sufficiently slowly to zero, which has not been previously studied.


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© Allerton Press, Inc. 2013

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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