Sum of Log Normal random variables

There is some folklore telling that scale-invariant distributions are build from the sum of lognormals. For instance, Huberman and Adamic. http://arxiv.org/abs/cond-mat/9901071 and http://arxiv.org/abs/cond-mat/0001459, based on the PhD thesis of Adamic. Also I would name Clauset, Rohilla Shalizi and Newman for a related topic, about how to distinguish between lognormal and scale-invariant specimens in the wild: http://arxiv.org/abs/0706.1062, http://arxiv.org/abs/cond-mat/0412004

Gao, Xu, Ye- Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables has many answers to your concern that was not addressed in Asmussen's paper.

you can see links for independent and correlated case:

http://airccj.org/CSCP/vol4/csit43104.pdf

http://airccj.org/CSCP/vol4/csit43105.pdf

Thanks