Sum of arbitrarily dependent random variables

Abstract

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 dependence structure.We show that under general tail conditions on two given distributions $F$ and $G$, there always exists a sequence of $F$-distributed random variables such that the scaled average of the sequence converges to a $G$-distributed limit almost surely. We construct such a sequence of random variables via a structure of conditional independence. The results in this paper suggest that with the common marginal distribution fixed and dependence structure unspecified, the distribution of the sum of a sequence of random variables can be asymptotically of any shape.