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2014 Sum of arbitrarily dependent random variables
Ruodu Wang
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Electron. J. Probab. 19: 1-18 (2014). DOI: 10.1214/EJP.v19-3373


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.


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Ruodu Wang. "Sum of arbitrarily dependent random variables." Electron. J. Probab. 19 1 - 18, 2014.


Accepted: 16 September 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1309.60029
MathSciNet: MR3263641
Digital Object Identifier: 10.1214/EJP.v19-3373

Primary: 60F15
Secondary: 60F05

Keywords: Almost sure convergence , arbitrary dependence , central limit theorems , laws of large numbers , regular variation

Vol.19 • 2014
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