Fully explicit large deviation inequalities for empirical processes with applications to information-based complexity

as a straightforward generalization of (1). To understand the convergence properties of αn it is a natural question to investigate the quantities (2) sup C∈C |αn(C)| and sup f∈F |αn(f)|, respectively. In the simple case of ξ1, ξ2, . . . being “ordinary” real random variables and C being the class of subintervals of R, a result of this type is the well-known Glivenko–Cantelli theorem. In the general case it turns out that the asymptotic order of the quantities in (2) depends on the