Distribution with a given moment generating function

As a follow-up to a question on a central limit theorem for independent random variables (r.v.) here, let $Y_j=-\log(1-V_j)$, where $V_j\sim\mbox{beta}(1-\sigma,j\sigma)$, $j\in\mathbb{N}^*$, $\sigma\in(0,1)$. The shifted sums $S_n=\sum_{j=1}^{n}Y_j -\frac{1-\sigma}{\sigma}\log n$ have moment generating functions (MGF) which admit a simple limit when $n\rightarrow \infty$: $$\mathbb{E}\left(e^{\lambda S_n}\right)\rightarrow M(\lambda)=\frac{\Gamma(1-\lambda/\sigma)}{\sigma^\lambda \Gamma(1-\lambda)}.$$

I'm trying to work out to the r.v. $S$ that admits $M(\lambda)$ as a MGF. The existence of $S$ is by the Kolmogorov three-series theorem, which ensures a.s. convergence. Note that $S$ is infinitely divisible since it is the limit of the infinitely divisible $Y_j$s.

In the expression of $M(\lambda)$, $\sigma^\lambda\Gamma(1-\lambda)$, resp. $\Gamma(1-\lambda/\sigma)$, is the MGF of a Gumbel r.v. shifted by $\log(\sigma)$, resp. Gumbel r.v. rescaled by $\sigma$. Though I don't see how to make use of this ratio since the inverse of an MGF isn't an MGF.