Suppose i have $n$ gamma random variables $X_i \sim \Gamma(\alpha_i,\beta_i)$, all with mgf (moment generating function) :

$$\varphi_i(t) = \mathbb{E}(e^{-tX_i}) = \left(\frac{\beta_i}{\beta_i - t}\right)^{\alpha_i}$$

Suppose furthermore that all thoose gamma random variables are comonotonous, meaning that, if we denote by $F_i$ the c.d.f for $X_i$, we have :

$$\forall i,j, X_j = F_j^{-1}\left(F_i(X_i)\right)$$

Can i express easily the distribution of the sum of thoose random variables ? I want the mgf, but if you get a density it's Ok too. To be clear, i want an expression for :

$$\varphi(t) = \mathbb{E}\left(e^{-t\sum\limits_{i=1}^{n}X_i}\right)$$

but my problem is that the c.d.f for gamma random variables writes :

$$F_i(x) = \frac{\gamma(\alpha_i,\beta_ix)}{\Gamma(\alpha_i)}$$

where $\gamma$ and $\Gamma$ are lower incomplete and complete gamma functions. The lower incomplete gamma function has no explicit inverse...


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