  We study $d$-variate problem in the average case setting with respect to a
zero-mean Gaussian measure. The covariance kernel of this Gaussian measure is a
product of univariate kernels and satisfies some special properties. We study
$(s, t)$-weak tractability of this multivariate problem, and obtain a necessary
and sufficient condition for $s>0$ and $t\in(0,1)$. Our result can apply to the
problems with covariance kernels corresponding to Euler and Wiener integrated
processes, Korobov kernels, and analytic Korobov kernels.
