We relax the moment conditions from a result in almost sure limit theory for U-statistics due to Berkes and Csaki (2001). We extend this result to the case of convergence to stable laws and also prove a functional version.

Let a class F of densities be given. We draw an i.i.d. sample from a density f which may or may not be in F . After every n, one must make a guess whether f 2 F or not. A class is almost surely testable if there exists such a testing sequence such that for any f , we make nitely many errors almost surely. In this paper, several results are given that allow one to decide whether a class is almos...

An integral analogue of the general almost sure limit theorem is presented. In the theorem, instead of a sequence of random elements, a continuous time random process is involved, moreover, instead of the logarithmical average, the integral of delta-measures is considered. Then the general theorem is applied to obtain almost sure versions of limit theorems for semistable and max-semistable proc...