EC-(s,t)-weak tractability of multivariate linear problems in the average case setting

We study EC-(s,t)-weak tractability of multivariate linear problems in the average case setting. This paper extends earlier work in the worst case setting. The parameters s≥0 and t≥0 allow us to study the information complexity n(ε,d) of a d-variate problem with respect to different powers of lnε⁻¹, corresponding to the bits of accuracy, and d. We consider the absolute and normalized error criteria. In particular, a multivariate problem is EC-(s,t)-weakly tractable iff lim_{d+ε⁻¹→∞}lnn(ε,d)∕[d^t+ln^sε⁻¹]=0. We deal with general linear problems and linear tensor product problems. We show necessary and sufficient conditions for EC-(s,t)-weak tractability. In the case of general linear problem these conditions are matching. For linear tensor product problems, we also show matching conditions with the exception of some cases where s>1, in general.