Convergence rates in precise asymptotics

Let X 1 , X 2 , … be i.i.d. random variables with partial sums S n , n ⩾ 1 . The now classical Baum–Katz theorem provides necessary and sufficient moment conditions for the convergence of ∑ n = 1 ∞ n r / p − 2 P ( | S n | ⩾ ε n 1 / p ) 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.