Approximation complexity of homogeneous sums of random processes

We study approximation properties of additive random fields $Y_d$, $d\in \mathbb{N}$, which are sums of zero-mean random processes with the same continuous covariance functions. The average case approximation complexity $n^{Y_d}(\epsilon )$ is defined as the minimal number of evaluations of arbitrary linear functionals needed to approximate $Y_d$, with relative $2$-average error not exceeding a given threshold $\epsilon \in (0,1)$. We investigate the growth of $n^{Y_d}(\epsilon )$ for arbitrary fixed $\epsilon \in (0,1)$ and $d\to \mathrm{\infty }$. The results are applied to sums of standard Wiener processes.