Electronic Journal of Probability

Sum of arbitrarily dependent random variables

Ruodu Wang

Full-text: Open access

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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065726

Digital Object Identifier
doi:10.1214/EJP.v19-3373

Mathematical Reviews number (MathSciNet)
MR3263641

Zentralblatt MATH identifier
1309.60029

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

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

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

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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