Suppose we have $X_1,...,X_n$ i.i.d rv's each with mgf $M_{X_i}(t)$.

Let $Z = \sum X_i$, we know that $M_Z(t) = M^{n}_{X_i}(t)$

In some cases, we can know the distribution of $Z$ by identifying the mgf of $Z$. In other cases is not possible to identify the mgf and the Jacobian method or any other alternative method is required.

Did anyone know under which family of distributions or what conditions are needed in order to obtain a "known" mgf for $Z$?

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    $\begingroup$ You may take a look at something like stable distribution / infinite divisibility. Those are some special classes of distribution and in mathematical terms, they are "closed under addition". So in such case you should be able to figure out the mgf / characteristic function of the sum. $\endgroup$ – BGM Apr 5 '17 at 5:21
  • $\begingroup$ @BGM take a look at my other question here $\endgroup$ – Richard Clare Apr 6 '17 at 16:33
  • $\begingroup$ I think here we have an algebraic structure. For example, we can define a subclass of distributions that the sum of the distribution can be evaluated using the moment generating functions IN the "closed under addition" distributions. Can we do the same for the product Z = XY ?. In general find the families of distributions closed under addition product convolution ? $\endgroup$ – Richard Clare Apr 6 '17 at 16:37

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