Journal of the
Korean Mathematical Society JKMS

In this paper, we obtain two general laws of precise asymptotics for sums of i.i.d random variables, which contain general weighted functions and boundary functions and also clearly show the relationship between the weighted functions and the boundary functions. As corollaries, we obtain Theorem 2 of Gut and Sp\v {a}taru [A. Gut and A. Sp\v {a}taru, Precise asymptotics in the law of the iterated logarithm, Ann. Probab. ${\boldsymbol 28}$ (2000), no. 4, 1870--1883] and Theorem 3 of Gut and Sp\v {a}taru [A. Gut and A. Sp\v {a}taru, Precise asymptotics in the Baum-Katz and Davids laws of large numbers, J. Math. Anal. Appl. ${\boldsymbol 248}$ (2000), 233--246].

Keywords: precise asymptotics, general law, weighted function, boundary function

MSC numbers: Primary 60F15; Secondary 60G50