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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.

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    $\begingroup$ 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). $\endgroup$
    – whuber
    Nov 19 at 14:03

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