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

TL;DR

This paper extends the Baum--Katz theorem to pairwise independent i.i.d. variables with general norming constants, under optimal moment conditions, using properties of slowly varying functions and advanced probabilistic techniques.

Contribution

It generalizes the Baum--Katz theorem to a broader class of sequences with pairwise independence and general norming, improving existing moment condition requirements.

Findings

Proves the Baum--Katz theorem for pairwise independent sequences

Uses properties of slowly varying functions and de Bruijn conjugates

Avoids maximal inequalities with Rio's techniques

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.