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

Consider a sequence of ramdom fields X _d , d ∈ $\mathbb{N}$, given by

$$ {X_d}(t)=\sum\limits_{{k\in {{\mathbb{N}}^d}}} {\prod\limits_{l=1}^d {\lambda \left( {{k_l}} \right){\xi_k}} \prod\limits_{l=1}^d } {\varphi_{{{k_l}}}}\left( {{t_l}} \right),\,\,\,\,\,\,t\in {{\left[ {0,1} \right]}^d}, $$

where ${{\left( {\lambda (i)} \right)}_{{ i\in \mathbb{N}}}}\in {l_2},\,\,{{\left( {{\varphi_i}} \right)}_{{i\in \mathbb{N}}}}$ is an orthonormal system in L ₂[0,1] , and ${{\left( {{\xi_k}} \right)}_{{k\in {{\mathbb{N}}^d}}}}$ are noncorrelated random variables with zero mean and unit variance. We study 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 → ∞. The result depends on the existence of a lattice structure of ${{\left( {\lambda (i)} \right)}_{{i\in \mathbb{N}}}}$. Bibliography: 9 titles.