caa a22 4500 605524629 CHVBK 20210128100754.0 cr unu---uuuuu 210128e20150101xx s 000 0 eng 10.1007/s10958-014-2195-2 doi (NATIONALLICENCE)springer-10.1007/s10958-014-2195-2 Khartov A. St. Petersburg State University, St. Petersburg, Russia aut Approximation in Probability of Tensor Product-Type Random Fields of Increasing Parametric Dimension [Elektronische Daten] [A. Khartov] We consider the sequence of Gaussian tensor product-type random fields Xd given by a multiparametric Karhunen-Loéve expansion X d t = ∑ k ∈ ℕ d ∏ l = 1 d λ k l 1 / 2 ξ k ∏ l = 1 d ψ k l t l , t ∈ 0 1 d , $$ {X}_d(t)={\displaystyle \sum_{k\in {\mathrm{\mathbb{N}}}^d}{\displaystyle \prod_{l=1}^d{\uplambda}_{k_l}^{1/2}{\xi}_k}{\displaystyle \prod_{l=1}^d{\psi}_{k_l}\left({t}_l\right),\kern0.36em t\in {\left[0,1\right]}^d},} $$ where (ξk)k∈ℕ dare standard Gaussian random variables and (λi)i∈ℕ and (ψi)i∈ℕ are the eigenvalues and eigenfunctions of the covariance operator of the process X1. We approximate Xd by finite sums X d (n) of the series using L2([0, 1]d)-norm ‖ ⋅ ‖ 2,d and study the exact asymptotics of the probabilistic approximation complexity n d p r ε δ : = min n ∈ ℕ : ℙ X d − X d n 2 , d 2 > ε 2 E X d 2 , d 2 ≤ δ $$ {n}_d^{\mathrm{pr}}\left(\varepsilon, \delta \right):= \min \left\{n\in \mathbb{N}:\mathrm{\mathbb{P}}\left({\left\Vert {X}_d-{X}_d^{(n)}\right\Vert}_{2,d}^2>{\varepsilon}^2\mathbb{E}{\left\Vert {X}_d\right\Vert}_{2,d}^2\right)\le \delta \right\} $$ in the case where the error threshold ε ∈ (0, 1) is fixed, the parametric dimension d→∞, and the confidence level δ = δd,ε may depend on ε and d. We show that under some conditions on (λi)i∈ℕ, the probabilistic complexity is asymptotically equivalent to the average approximation complexity n d a v g ε : = min n ∈ ℕ : E X d − X d n 2 , d 2 ≤ ε 2 E X d 2 , d 2 $$ {n}_d^{\mathrm{avg}}\left(\varepsilon \right):= \min \left\{n\in \mathbb{N}:\mathbb{E}{\left\Vert {X}_d-{X}_d^{(n)}\right\Vert}_{2,d}^2\le {\varepsilon}^2\mathbb{E}{\left\Vert {X}_d\right\Vert}_{2,d}^2\right\} $$ The result depends on the existence of a lattice structure of the sequence (λi)i∈ℕ. Bibliography: 10 titles. Springer Science+Business Media New York, 2014 Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 204/1(2015-01-01), 165-179 1072-3374 204:1<165 2015 204 10958 https://doi.org/10.1007/s10958-014-2195-2 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10958-014-2195-2 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Khartov A. St. Petersburg State University, St. Petersburg, Russia aut NATIONALLICENCE 773 0- Journal of Mathematical Sciences Springer US; http://www.springer-ny.com 204/1(2015-01-01), 165-179 1072-3374 204:1<165 2015 204 10958