Consider a random variable $S$ defined as the sum of independent random variables:
${\displaystyle {S = X_{1} + X_{2} + ... + X_{n}}}$
As $n$ approaches infinity, the prob density of $S$ is given by $f(s)$ for $0 < s < {\infty}$.
We aim to obtain a prob density $g(t)$ such that:
${\displaystyle {g(t) = f(e^t)e^t}， {-\infty} < t < {+\infty}}$.
Question:
How can we adjust the random variable for each step $X_{i}$ to $\hat X_{i}$ or redefine (or modify) $S$, to achieve the desired prob density $g(t)$? We can assume the prob density of $X_{i}$ is $h_{i}(x)$, they belong to the same family of Gamma distribution, but with different parameters.