Approximation in probability of tensor product-type random fields of increasing parametric dimension

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.