General laws of precise asymptotics for sums of random variables
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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 Spataru [A. Gut and A. Spataru,Precise asymptotics in the law of the iterated logarithm, Ann. Probab. 28 (2000), no. 4, 1870{1883] and Theorem 3 of Gut and Spataru [A. Gut and A. Spataru, Precise asymptotics in the Baum-Katz and Davids laws of large numbers, J. Math. Anal. Appl. 248 (2000), 233{246].
Abstract
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 Spataru [A. Gut and A. Spataru,Precise asymptotics in the law of the iterated logarithm, Ann. Probab. 28 (2000), no. 4, 1870{1883] and Theorem 3 of Gut and Spataru [A. Gut and A. Spataru, Precise asymptotics in the Baum-Katz and Davids laws of large numbers, J. Math. Anal. Appl. 248 (2000), 233{246].
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