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

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 $(\lambda(i))_{i\in\mathbb N}$.