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Hi,

I would be gratefull 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

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3 Answers 3

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

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:-( I have reviewed some of the references, and another bunch of pdf, and it seems they are only interested on summations with an exponential weight, so it is unclear how general the folk-lore is. –  arivero Aug 27 '10 at 15:02
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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.

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up vote 0 down vote accepted

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

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It sounds like the article by Asmussen would be exactly what you're looking for. Have you looked at its cited-by on Google Scholar, or contacted the author directly? scholar.google.com/scholar?cites=9410380893780005190 –  Tom LaGatta Jul 6 '10 at 19:40
    
Hi Tom, It is a very interesting article indeed and at first I thought it would solve all my problems. But it only treats the right tail and says nothing about the left tail which is of equal interest to me. Another point is that I would be interested in the approximation of density and I don't think it is licit to approximate $dP(S_n>x)/dx$ using the derivatitve of an asymptotic equivalent of $P(S_n>x)$ (or at least I can't see under what condition this holds true). Finally no I didn't contact the authors but maybe I will. Regards –  The Bridge Jul 7 '10 at 8:21
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