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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?".


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))

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closed as unclear what you're asking by Yemon Choi, Ian Morris, Suvrit, Carlo Beenakker, Jeremy Rouse Jan 23 '15 at 1:08

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question.If this question can be reworded to fit the rules in the help center, please edit the question.

up vote 4 down vote accepted

you can see links for independent and correlated case:


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Can you add a little description of the contents of these pdf documents so that this answer remains meaningful even if the links die? – Joonas Ilmavirta Jan 15 '15 at 6:26
@ Mawane : Excellent references thank you. I'm marking your answer as accepted. Best regards – The Bridge Jan 15 '15 at 22:16

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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There is some folklore telling that scale-invariant distributions are build from the sum of lognormals. For instance, Huberman and Adamic. and, 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:,

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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


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? – 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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