On the Baum--Katz theorem for sequences of pairwise independent random variables with regularly varying normalizing constants

Abstract

This paper proves the Baum--Katz theorem for sequences of pairwise independent identically distributed random variables with general norming constants under optimal moment conditions. The proof exploits some properties of slowly varying functions and the de Bruijn conjugates, and uses the techniques developed by Rio (1995) to avoid using the maximal type inequalities.

Comptes Rendus Mathématique, 358, 2020, 1231--1238. A typo at the last line of equation (34) in the journal version is corrected