Tractability of tensor product problems in the average case setting

It has been an open problem to derive a necessary and sufficient condition for a linear tensor product problem S = {Sd} in the average case setting to be weakly tractable but not polynomially tractable. As a result of the tensor product structure, the eigenvalues of the covariance operator of the induced measure in the one dimensional problem characterize the complexity of approximating Sd, d ≥ 1, with accuracy ε. If ∑∞ j=1 λj < 1 and λ2 > 0, we know that S is not polynomially tractable iff lim supj→∞ λjj = ∞ for all p > 1. Thus we settle the open problem by showing that S is weakly tractable iff ∑ j>n λj = o ( ln−2 n ) . In particular, assume that ` = lim j→∞ λjj ln(j + 1), exists. Then S is weakly tractable iff ` = 0.