Approximation in Probability of Tensor Product-Type Random Fields of Increasing Parametric Dimension
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| 024 | 7 | 0 | |a 10.1007/s10958-014-2195-2 |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s10958-014-2195-2 | ||
| 100 | 1 | |a Khartov |D A. |u St. Petersburg State University, St. Petersburg, Russia |4 aut | |
| 245 | 1 | 0 | |a Approximation in Probability of Tensor Product-Type Random Fields of Increasing Parametric Dimension |h [Elektronische Daten] |c [A. Khartov] |
| 520 | 3 | |a 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. | |
| 540 | |a Springer Science+Business Media New York, 2014 | ||
| 773 | 0 | |t Journal of Mathematical Sciences |d Springer US; http://www.springer-ny.com |g 204/1(2015-01-01), 165-179 |x 1072-3374 |q 204:1<165 |1 2015 |2 204 |o 10958 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s10958-014-2195-2 |q text/html |z Onlinezugriff via DOI |
| 898 | |a BK010053 |b XK010053 |c XK010000 | ||
| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s10958-014-2195-2 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 100 |E 1- |a Khartov |D A. |u St. Petersburg State University, St. Petersburg, Russia |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Journal of Mathematical Sciences |d Springer US; http://www.springer-ny.com |g 204/1(2015-01-01), 165-179 |x 1072-3374 |q 204:1<165 |1 2015 |2 204 |o 10958 | ||