In many classic problems of asymptotic analysis, it appears that the scaled average of a sequence of F -distributed random variables converges to G-distributed limit in some sense of convergence. In this paper, we look at the classic convergence problems from a novel perspective: we aim to characterize all possible limits of the sum of a sequence of random variables under different choices of d...

Let {Xn, n ≥ 1} be a sequence of i.i.d., real random variables. Hsu and Rabbins [5] proved that if E(X) = 0 and E(X) < ∞, then the sequence 1 n ∑n i=1 Xi converges to 0 completely. (i.e., the series ∑∞ n=1 P [|Sn| > nε] < ∞, converges for every ε > 0). Now let {Xn, n ≥ 1} be a sequence of negatively dependent real random variables. In this paper, we proved the complete convergence of the sequen...