Convergence rates in precise asymptotics II

Abstract. Let \(X_1,\, X_2,\,\dots\) be independent, identically distributed (i.i.d.) random variables with partial sums \(S_n\), \(n\geq1\). The now classical Baum–Katz problem concerns finding necessary and sufficient moment conditions for the convergence of \(\sum_{n=1}^\infty n^{(r/p)-2}P(|S_n| \geq \varepsilon n^{1/p})\) for fixed \(\varepsilon>0\). A now equally classical paper by Heyde in 1975 initiated what has later been called precise asymptotics, namely asymptotics for the same sum (for the case \(r=2\) and \(p=1\)) when, instead, \(\varepsilon\searrow0\). In a predecessor of this paper we extended a result due to Klesov (1994), in which he determined the convergence rate in Heyde's theorem, to the case \(r\geq2\), \(0 <p <2\). The present companion paper is devoted to the case when the summands belong to the normal domain of attraction of a stable distribution with index \(\alpha\in(1,2]\), in particular to the analog related to Spitzer's 1956-theorem.