Average Case $(s,t)$-weak tractability of non-homogenous tensor product problems

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.