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JOURNALS // Zapiski Nauchnykh Seminarov POMI // Archive

Zap. Nauchn. Sem. POMI, 2011 Volume 396, Pages 233–256 (Mi znsl4664)

This article is cited in 2 papers

Average approximation of tensor product-type random fields of increasing dimension

A. A. Khartov

Saint-Petersburg State University, Saint-Petersburg, Russia

Abstract: 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 $(\lambda(i))_{i\in\mathbb N}$.

Key words and phrases: tensor product-type random fields, average approximation, average-case complexity of approximation, curse of dimensionality, exact asymptotic behavior.

UDC: 519.21

Received: 21.10.2011


 English version:
Journal of Mathematical Sciences (New York), 2013, 188:6, 769–782

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