Precise asymptotics in Spitzer and Baum–Katz’s law of large numbers: the semistable case

Let X1,X2, . . . be i.i.d. random variables with distribution μ and with mean zero, whenever the mean exists. Set Sn = X1 +···+Xn. In recent years precise asymptotics as ε ↓ 0 have been proved for sums like ∞n=1 n−1P{|Sn| εn1/p}, assuming that μ belongs to the (normal) domain of attraction of a stable law. Our main results generalize these results to distributions μ belonging to the (normal) domain of semistable attraction of a semistable law. Furthermore, a limiting case new even in the stable situation is presented.  2003 Elsevier Inc. All rights reserved