Random Variables ($) Arrays Of Full Convergence

In the paper, we discuss the convergence equivalent of the weighted sums for B-valued random element array, the complete convergence of the weighted sums for B-valued random element array, the complete convergence of the weighted sums for ND random variable array and the complete convergence for difference distribution dependent random variable sequence.In the first chapter, we find a nice sufficient conditions for strong law of large numbers without the restrict conditions on Banach space geometry properties. Then we also establish some equivalent relatio- ns between probability convergence, almost surely convergence and complete convergence .These works weaken the conditions for the results of Cabrera and Sung (2000) and Anna Kuczmaszewka,Dominik Szynal (2003), and extending their results to the weighted sums of the random array, perfecting the foundation conclusions of equivalent relations of random element sequence.In the second chapter, on the foundations of the recent complete convergence for the B-valued random variable sequence and the work made by Wang, Rao and Yang (1993), we establish some com- plete convergence theorems for the weighted sums of the B-valued random element sequence. We also improve Liang et al (1997) results, and extend those results to random element array.