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-Liebler 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.