On convergence of empirical point processes

A new approach to the study of asymptotic behavior of truncated sums1bn∑i⩽n,i∈AnXiis proposed, where (An)↑(An)↑ is an increasing sequence of sets and (Xi)(Xi) are i.i.d. random vectors from the domain of attraction of a stable law. This approach is based on the investigation of weak convergence (more strong than usual) for empirical point processes associated with the vectors (Xi).(Xi).