Of Weighted Sums Of Independent Random Variables And Iterated Logarithm Law

The Law of Iterated Logarithm for independent r.v. has been deeply discussed in all kinds of paper. We have been familiar with "the Law of Iterated Logarithm of Kolmogorov" and "the Law of Iterated Logarithm of Hartman-Wintner".This paper will mainly discuss the Law of Iterated Logarithm for some kind weighted partial sum. It is an extension of "the Law of Iterated Logarithm of Kolmogorov".In the course of proving,we extend two lemmas and use the relating results of partial sum increment of Independent r.v. As for i.i.d.r.v., we get the extension of "the Law of Iterated Logarithm of Hartman-Wintner" under weaker conditions. At the end of this paper,we discuss that the moment conditions of theorem are necessary to the Law of Iterated Logarithm of this form .This paper has three parts , Introduction gives the origin of the problem,some definitions and some lemmas. In the second chapter,we give the proof of extended lemmas and theorems in details.In the third chapter ,we discuss the necessity of moment conditions in the case of i.i.d.r.v..