Electronic Journal of Probability

Sum of arbitrarily dependent random variables

Ruodu Wang

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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.

Article information

Electron. J. Probab., Volume 19 (2014), paper no. 84, 18 pp.

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

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F15: Strong theorems
Secondary: 60F05: Central limit and other weak theorems

central limit theorems laws of large numbers almost sure convergence arbitrary dependence regular variation

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Wang, Ruodu. Sum of arbitrarily dependent random variables. Electron. J. Probab. 19 (2014), paper no. 84, 18 pp. doi:10.1214/EJP.v19-3373. https://projecteuclid.org/euclid.ejp/1465065726

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