@article{EJP3373,
author = {Ruodu Wang},
title = {Sum of arbitrarily dependent random variables},
journal = {Electron. J. Probab.},
fjournal = {Electronic Journal of Probability},
volume = {19},
year = {2014},
keywords = {central limit theorems; laws of large numbers; almost sure convergence; arbitrary dependence; regular variation},
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.},
pages = {no. 83, 1-18},
issn = {1083-6489},
doi = {10.1214/EJP.v19-3373},
url = {http://ejp.ejpecp.org/article/view/3373}}