Geodesic
Approximation in probability of tensor product-type random fields of increasing parametric dimension
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 19, Tome 412 (2013), pp. 252-273 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Consider a sequence of Gaussian tensor product-type random fields $X_d$, $d\in\mathbb N$, given by $$ X_d(t)=\sum_{k\in\widetilde{\mathbb N}^d}\prod_{l=1}^d\lambda_{k_l}^{1/2}\,\xi_k\,\prod_{l=1}^d\psi_{k_l}(t_l),\quad t\in [0,1]^d, $$ where $(\lambda_i)_{i\in\widetilde{\mathbb N}}$ and $(\psi_i)_{i\in\widetilde{\mathbb N}}$ are all positive eigenvalues and eigenfunctions of covariance operator of process $X_1$, $(\xi_k)_{k\in\widetilde{\mathbb N}}$ are standard Gaussian random variables, and $\widetilde{\mathbb N}$ is a subset of natural numbers. We investigate the exact asymptotic behavior of probabilistic complexity of approximation for $X_d$ by partial sums $X_d^{(n)}$: $$ n_d^{pr}(\varepsilon,\delta):=\min\Bigl\{n\in\mathbb N\colon\mathbf P\left(\|X_d-X_d^{(n)}\|^2_{2,d}>\varepsilon^2 \,\mathbf E\|X_d\|^2_{2,d}\right)\leqslant\delta\Bigr\}, $$ when the parametric dimension $d\to\infty$, the error threshold $\varepsilon\in(0,1)$ is fixed, and the confidence level $\delta=\delta_{d,\varepsilon}$ may go to zero. 
@article{ZNSL_2013_412_a13,
     author = {A. A. Khartov},
     title = {Approximation in probability of tensor product-type random fields of increasing parametric dimension},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {252--273},
     year = {2013},
     volume = {412},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2013_412_a13/}
}
TY  - JOUR
AU  - A. A. Khartov
TI  - Approximation in probability of tensor product-type random fields of increasing parametric dimension
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2013
SP  - 252
EP  - 273
VL  - 412
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2013_412_a13/
LA  - ru
ID  - ZNSL_2013_412_a13
ER  - 
%0 Journal Article
%A A. A. Khartov
%T Approximation in probability of tensor product-type random fields of increasing parametric dimension
%J Zapiski Nauchnykh Seminarov POMI
%D 2013
%P 252-273
%V 412
%U http://geodesic.mathdoc.fr/item/ZNSL_2013_412_a13/
%G ru
%F ZNSL_2013_412_a13
A. A. Khartov. Approximation in probability of tensor product-type random fields of increasing parametric dimension. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 19, Tome 412 (2013), pp. 252-273. http://geodesic.mathdoc.fr/item/ZNSL_2013_412_a13/

[1] R. Adler, J. Taylor, Random Fields and Geometry, Springer, New York, 2007 | MR

[2] C.-G. Esseen, “Fourier analysis of distribution functions. A mathematical study of the Laplace–Gaussian law”, Acta Math., 77 (1945), 1–125 | DOI | MR | Zbl

[3] M. A. Lifshits, E. V. Tulyakova, “Curse of dimentionality in approximation of random fields”, Probab. Math. Stat., 26:1 (2006), 97–112 | MR | Zbl

[4] E. Novak, I. H. Sloan, J. F. Traub, H. Wózniakowski, Essays on the Complexity of Continuous Problems, European Mathematical Society Publishing House, Zürich, 2009 | MR | Zbl

[5] E. Novak, H. Wózniakowski, Tractability of Multivariate Problems, v. I, EMS Tracts Math., 6, Linear Information, European Mathematical Society Publishing House, Zürich, 2008 | MR | Zbl

[6] E. Novak, H. Wózniakowski, Tractability of Multivariate Problems, v. II, EMS Tracts Math., 12, Standard Information for Functionals, European Mathematical Society Publishing House, Zürich, 2010 | MR | Zbl

[7] E. Novak, H. Wózniakowski, Tractability of Multivariate Problems, v. III, EMS Tracts Math., 18, Standard Information for Operators, European Mathematical Society Publishing House, Zürich, 2012 | MR | Zbl

[8] K. Ritter, Average-case Analysis of Numerical Problems, Lect. Notes Math., 1733, Springer, Berlin, 2000 | DOI | MR | Zbl

[9] V. V. Petrov, Predelnye teoremy dlya summ nezavisimykh sluchainykh velichin, Nauka, M., 1987 | MR

[10] A. A. Khartov, “Approksimatsiya v srednem tenzornykh sluchainykh polei vozrastayuschei razmernosti”, Zap. nauchn. semin. POMI, 396, 2011, 233–256 | MR