Convergence rates in precise asymptotics II
Allan Gut, Josef Steinebach · Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae Sectio computatorica · 2013
Let X1, X2, . . .be independent, identically distributed (i.i.d.) random variables with partial sums Sn, n ≥ 1.The now classical Baum-Katz problem concerns finding necessary and sufficient moment conditions for the convergence of ∞ n=1 n (r/p)-2 P (|Sn| ≥ εn 1/p ) for fixed ε > 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, ε 0. 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 ≥ 2, 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 α ∈ (1, 2], in particular to the analog related to Spitzer's 1956theorem.