International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 630857, 28 pages
doi:10.1155/2009/630857
Research Article

Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables

Xin Gao,1 Hong Xu,1 and Dong Ye2

1Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto, ON, M3J 1P3, Canada
2LMAM, Université de Metz, UFR 7122 Bâtiment A, Île de Saulcy, 57045 Metz Cedex 1, France

Received 8 August 2008; Revised 3 February 2009; Accepted 5 March 2009

Academic Editor: Michael Evans

Copyright © 2009 Xin Gao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider the asymptotic behavior of a probability density function for the sum of any two lognormally distributed random variables that are nontrivially correlated. We show that both the left and right tails can be approximated by some simple functions. Furthermore, the same techniques are applied to determine the tail probability density function for a ratio statistic, and for a sum with more than two lognormally distributed random variables under some stricter conditions. The results yield new insights into the problem of characterization for a sum of lognormally distributed random variables and demonstrate that there is a need to revisit many existing approximation methods.