## Electronic Journal of Probability

### Sum of arbitrarily dependent random variables

Ruodu Wang

#### 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

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

Rights

#### 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

#### References

• Bernard, Carole; Jiang, Xiao; Wang, Ruodu. Risk aggregation with dependence uncertainty. Insurance Math. Econom. 54 (2014), 93–108.
• Chaganty, N. Rao; Joe, Harry. Range of correlation matrices for dependent Bernoulli random variables. Biometrika 93 (2006), no. 1, 197–206.
• Distributions with given marginals and statistical modelling. Papers from the meeting held in Barcelona, July 17-20, 2000. Edited by Carles M. Cuadras, Josep Fortiana and JosÃ© A. Rodriguez-Lallena. Kluwer Academic Publishers, Dordrecht, 2002. xxiv+244 pp. ISBN: 1-4020-0914-3
• Advances in probability distributions with given marginals. Beyond the copulas. Papers from the Symposium on Distributions with Given Marginals held in Rome, April 1990. Edited by G. Dall'Aglio, S. Kotz and G. Salinetti. Mathematics and its Applications, 67. Kluwer Academic Publishers Group, Dordrecht, 1991. xviii+231 pp. ISBN: 0-7923-1156-6
• de Haan, Laurens; Ferreira, Ana. Extreme value theory. An introduction. Springer Series in Operations Research and Financial Engineering. Springer, New York, 2006. xviii+417 pp. ISBN: 978-0-387-23946-0; 0-387-23946-4
• Embrechts, Paul; Puccetti, Giovanni. Bounds for functions of multivariate risks. J. Multivariate Anal. 97 (2006), no. 2, 526–547.
• Embrechts, P., G. Puccetti, and L. Rüschendorf (2013). Model uncertainty and VaR aggregation. J. Bank. Financ./~ 37/(8), 2750–2764.
• Embrechts, P., G. Puccetti, L. Rüschendorf, R. Wang, and A. Beleraj (2014). An academic response to Basel 3.5. phRisks, 2(1), 25–48.
• Fréchet, Maurice. Sur les tableaux de corrélation dont les marges sont données. (French) Ann. Univ. Lyon. Sect. A. (3) 14, (1951). 53–77.
• Hoeffding, Wassilij. Maszstabinvariante Korrelationstheorie. (German) Schr. Math. Inst. u. Inst. Angew. Math. Univ. Berlin 5, (1940). 181–233.
• Joe, Harry. Multivariate models and dependence concepts. Monographs on Statistics and Applied Probability, 73. Chapman & Hall, London, 1997. xviii+399 pp. ISBN: 0-412-07331-5
• Makarov, G. D. Estimates for the distribution function of the sum of two random variables with given marginal distributions. (Russian) Teor. Veroyatnost. i Primenen. 26 (1981), no. 4, 815–817.
• McNeil, Alexander J.; Frey, Rüdiger; Embrechts, Paul. Quantitative risk management. Concepts, techniques and tools. Princeton Series in Finance. Princeton University Press, Princeton, NJ, 2005. xvi+538 pp. ISBN: 0-691-12255-5
• Nelsen, Roger B. An introduction to copulas. Second edition. Springer Series in Statistics. Springer, New York, 2006. xiv+269 pp. ISBN: 978-0387-28659-4; 0-387-28659-4
• Rüschendorf, Ludger. Random variables with maximum sums. Adv. in Appl. Probab. 14 (1982), no. 3, 623–632.
• Rüschendorf, Ludger. Mathematical risk analysis. Dependence, risk bounds, optimal allocations and portfolios. Springer Series in Operations Research and Financial Engineering. Springer, Heidelberg, 2013. xii+408 pp. ISBN: 978-3-642-33589-1; 978-3-642-33590-7
• Shaked, Moshe; Shanthikumar, J. George. Stochastic orders. Springer Series in Statistics. Springer, New York, 2007. xvi+473 pp. ISBN: 978-0-387-32915-4; 0-387-32915-3.
• Tchen, André H. Inequalities for distributions with given marginals. Ann. Probab. 8 (1980), no. 4, 814–827.
• Wang, Bin; Wang, Ruodu. The complete mixability and convex minimization problems with monotone marginal densities. J. Multivariate Anal. 102 (2011), no. 10, 1344–1360.
• Wang, Ruodu; Peng, Liang; Yang, Jingping. Bounds for the sum of dependent risks and worst value-at-risk with monotone marginal densities. Finance Stoch. 17 (2013), no. 2, 395–417.