  We study approximation properties of sequences of centered additive random
fields $Y_d$, $d\in\mathbb{N}$. The average case approximation complexity
$n^{Y_d}(\varepsilon)$ is defined as the minimal number of evaluations of
arbitrary linear functionals that is needed to approximate $Y_d$ with relative
$2$-average error not exceeding a given threshold $\varepsilon\in(0,1)$. We
investigate the growth of $n^{Y_d}(\varepsilon)$ for arbitrary fixed
$\varepsilon\in(0,1)$ and $d\to\infty$. Under natural assumptions we obtain
general results concerning asymptotics of $n^{Y_d}(\varepsilon)$. We apply our
results to additive random fields with marginal random processes corresponding
to the Korobov kernels.
