Asymptotic behavior of tail density for sum of correlated lognormal variables

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.