On complete convergence of triangular arrays of independent random variables

Given a triangular array a={an,k,1⩽k⩽kn,n⩾1} of positive reals, we study the complete convergence property of Tn=∑k=1knan,kXn,k for triangular arrays X={Xn,k,1⩽k⩽kn,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 Xn,k are in Lp, and establish a link between the Lp-case and L2p-case in terms of densities. We finally obtain a density type condition in the case of uniformly bounded random variables.