On complete convergence of triangular arrays of independent random variables

Abstract

Given a triangular array a = { a n , k , 1 ⩽ k ⩽ k n , n ⩾ 1 } of positive reals, we study the complete convergence property of T n = ∑ k = 1 k n a n , k X n , k for triangular arrays X = { X n , k , 1 ⩽ k ⩽ k n , n ⩾ 1 } of independent random variables. In the Gaussian case we obtain a simple characterization of density type. Using Skorohod representation and Gaussian randomization, we then derive sufficient criteria for the case when X n , k are in L p , and establish a link between the L p -case and L 2 p -case in terms of densities. We finally obtain a density type condition in the case of uniformly bounded random variables.