Complete Convergence for M-Pairwise Negatively Dependent Random Variables

Abstract

Hsu and Robbins (1947) introduce the concept complete convergence as follows. A sequence of random variables is said to converge completely to a constant if for all The converse is true if the are independent. They also show that the sequence of arithmetic means of independent and identically distributed random variables converges completely to the expected value if the variance of the summands is finite. Erdös proved the converse. The result of Hsu-Robbins-Erdös is a fundamental theorem in probability theory and has been generalized and extended in several directions by many authors. In this paper, let be a sequence of positive constants with and be a sequence of m-pairwise negatively dependent random variables. We study the complete convergence for m-pairwise negatively dependent random variables under mild condition . Our results obtained in the paper generalize the corresponding ones for pairwise independent and identically distributed random variables and also pairwise negatively dependent random variables.