Convergence rates in precise asymptotics

Let X1,X2,… be i.i.d. random variables with partial sums Sn, n⩾1. The now classical Baum–Katz theorem provides necessary and sufficient moment conditions for the convergence of for fixed ε>0. An equally classical paper by Heyde in 1975 initiated what is now called precise asymptotics, namely asymptotics for the same sum (for the case r=2 and p=1) when, instead, ε↘0. In this paper we extend a result due to Klesov (1994), in which he determined the convergence rate in Heydeʼs theorem.