Geodesic
Average approximation of tensor product-type random fields of increasing dimension
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 17, Tome 396 (2011), pp. 233-256 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

Consider a sequence of random fields $X_d$, $d\in\mathbb N$, given by $$ X_d(t)=\sum_{k\in\mathbb N^d}\prod^d_{l=1}\lambda(k_l)\xi_k\prod^d_{l=1}\varphi_{k_l}(t_l),\quad t\in[0,1]^d, $$ where $(\lambda(i))_{i\in\mathbb N}\in l_2$, $(\varphi_i)_{i\in\mathbb N}$ is an orthonormal system in $L_2[0,1]$ and $(\xi_k)_{k\in\mathbb N^d}$ are non-correlated random variables with zero mean and unit variance. We investigate the exact asymptotic behavior of average-case complexity of approximation to $X_d$ by $n$-term partial sums providing a fixed level of relative error, as $d\to\infty$. The result depends on existence of lattice structure of $(\lambda(i))_{i\in\mathbb N}$. 
@article{ZNSL_2011_396_a16,
     author = {A. A. Khartov},
     title = {Average approximation of tensor product-type random fields of increasing dimension},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {233--256},
     year = {2011},
     volume = {396},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2011_396_a16/}
}
TY  - JOUR
AU  - A. A. Khartov
TI  - Average approximation of tensor product-type random fields of increasing dimension
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2011
SP  - 233
EP  - 256
VL  - 396
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2011_396_a16/
LA  - ru
ID  - ZNSL_2011_396_a16
ER  - 
%0 Journal Article
%A A. A. Khartov
%T Average approximation of tensor product-type random fields of increasing dimension
%J Zapiski Nauchnykh Seminarov POMI
%D 2011
%P 233-256
%V 396
%U http://geodesic.mathdoc.fr/item/ZNSL_2011_396_a16/
%G ru
%F ZNSL_2011_396_a16
A. A. Khartov. Average approximation of tensor product-type random fields of increasing dimension. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 17, Tome 396 (2011), pp. 233-256. http://geodesic.mathdoc.fr/item/ZNSL_2011_396_a16/

[1] G. Vasilkovskii, G. Voznyakovskii, “Obzor slozhnosti v srednei situatsii dlya lineinykh mnogomernykh problem”, Izv. vuzov. Matem., 2009, no. 4, 3–19 | MR

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

[3] N. A. Serdyukova, “Zavisimost slozhnosti approksimatsii sluchainykh polei ot razmernosti”, Teoriya veroyatn. i ee primen., 54:2 (2009), 256–270 | DOI | MR | Zbl

[4] 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

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

[6] A. Papageorgiou, I. Petras, “Tractability of tensor product problems in the average case setting”, J. Complexity, 27:3–4 (2011), 273–280 | DOI | MR | Zbl

[7] 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

[8] E. Novak, H. Wózniakowski, Tractability of multivariate problems, v. II, EMS Tracts Math., 12, Standart Information for Functionals, European Mathematical Society Publishing House, Zürich, 2010 | MR | Zbl

[9] 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