Generalized Law of the Iterated Logarithm and Its Convergence Rate

Abstract

This article considers the partial sums from a sequence of independent and identically distributed random variables. It is well-known that the Hartman-Wintner law of the iterated logarithm holds if and only if the second moment exists. This article studies the generalized law of the iterated logarithm for the partial sums when they are normalized by a sequence of constants that are regularly varying with index 1/2. As a result, two equivalent conditions for the law are obtained.

Keywords:

• Complete convergence
• Independent random variables
• Laws of the iterated logarithm
• Sums

Mathematics Subject Classification (2000):

• 60F15
• 60G50

Acknowledgments

The authors thank the referee for a careful reading of the manuscript and helpful comments. Chen's research was supported by the National Nature Science Foundation of China, and Qi's research was partially supported by NSF grant DMS-0604176.