  We study approximation properties of sequences of centered random elements
$X_d$, $d\in\mathbb{N}$, with values in separable Hilbert spaces. We focus on
sequences of tensor product-type random elements, which have covariance
operators of corresponding tensor product form. The average case approximation
complexity $n^{X_d}(\varepsilon)$ is defined as the minimal number of
evaluations of arbitrary linear functionals that is needed to approximate $X_d$
with relative $2$-average error not exceeding a given threshold
$\varepsilon\in(0,1)$. The growth of $n^{X_d}(\varepsilon)$ as a function of
$\varepsilon^{-1}$ and $d$ determines whether a sequence of corresponding
approximation problems for $X_d$, $d\in\mathbb{N}$, is tractable or not.
Different types of tractability were studied in the paper by M. A. Lifshits, A.
Papageorgiou and H. Wo\'zniakowski (2012), where for each type the necessary
and sufficient conditions were found in terms of the eigenvalues of the
marginal covariance operators. We revise the criterion of quasi-polynomial
tractability and provide its simplified version. We illustrate our result by
applying to random elements corresponding to tensor products of squared
exponential kernels. Also we extend recent result of G. Xu (2014) concerning
weighted Korobov kernels.
