Sum of Log Normal random variables [closed]

Question

I would be grateful to anyone who could provide me with some reference concerning the behavior of the sum of Log Normal random variables (need not independent) with respect to a Log Normal random variable.

It is obvious that such a sum has no reason to be log normal but what is less obvious to me is "is it very far from a log normal distribution really ? " or "under what conditions those two objects could be considered as close or very close?".

Regards

Edit : A note to explain the "general" formulation of the question :

There are many metrics to evaluate distance between 2 random variables (Kullback-Leibler and entropic-like metrics, total variations, Hellinger, and so on), I have asked the question in this unspecified fashion because I had no "a priori" on the metric or any other indicator that could cope with the subject, on the contrary the more approaches I could get, the happier. Second aspect of the question, which is implicit in its general formulation, is the way in which a sum of log-normal could be approximated by a log-normal variable. At the time, I had no insight on the methodology to do so. In respect to this second aspect I still think that a general formulation of the question is for the better. Nevertheless I have extended the tags with "reference request" so that it is more clear to the reader I'd rather be pointed to relevant literature on the subject than get a direct answer in a post. ((spelling of "Kullback-Leibler" corrected, is seen wrong, too often))

Answer 1

you can see links for independent and correlated case:

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

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

Thanks

Answer 2

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

Answer 3

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.

Answer 4

Hi,

Here are the most intersting reference I could find

I post them because I think other people might be intereted in this issue, so here are the articles titles :

Asmussen, Rojas-Nandayapa - Sums of Dependent lognormal Random Variables, Asymptotics and Simulation

Vanduffel, Chen, Dhaene, Goovaerts, Henrard - Optimal Approximations for Risk Measures of Sums of Lognormals based on Conditional Expectations

Li - A Novel Accurate Approximation Method of Lognormal Sum Random Variables

Gao, Xu, Ye- Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables

Mehta, Molisch, Wu, Zhang - Approximating the Sum of Correlated Lognormal or Lognormal-Rice Random Variables

Fu, Madan, Wang - Pricing Continuous Asian Options, A Comparison of Monte Carlo and Laplace Transform Inversion Methods

Vecer - New Pricing of Asian Options

And a few other articles not directly applicable to this problem but interesting on their own :

Eden, Viens - General Upper and Lower Tail Estimates using Malliavin Calculus and Stein's Equations

Barndorf-Nielsen, Kluppelberg - Tail Exactness of Multivariate Saddlepoint Approximations

Have a nice day