Recurrence for Products of Renewal Sequences

Abstract

If $(u_n)^\infty_{n=0}$ is a null-recurrent renewal sequence, we prove that there exist two null-recurrent renewal sequences $(\nu_n)^\infty_{n=0}$ and $(w_n)^\infty_{n=0}$ such that $(u_n\nu_n)^\infty_{n=0}$ is null-recurrent and $(u_n w_n)^\infty_{n=0}$ is transient.