# Distribution of sum of independent but not i.i.d. lognormal variables?

I am trying to find the distribution of the following variable Z:

$$X_i$$ are each independent with Lognormal distribution ($$\mu_i, \sigma^2_i$$), $$X_i \in L^2$$ forall $$\forall i$$

Z = $$\sum_i cX_i$$ where $$c$$ is some non-negative scalar

The Fenton-Wilkinson approximation does not work since the r.v.s are not i.i.d and thus I'm not sure how to find the distribution so I can compute probabilities.

• There is no closed form. Sometimes, geostatisticians invoke a "principle of conservation of lognormality" to approximate such linear combinations with a lognormal distribution, knowing YMMV. The value of $c$ is irrelevant, btw, because it can be absorbed in all the $\mu_i.$ The shape of the sum is determined primarily by how much the $\sigma_i$ vary and whether any of them are extreme (watch out for values greater than about 0.6, typically).
– whuber
Nov 19 at 14:03