If $F(s)$ is the Laplace transform of $f(t)$ and \begin{equation} F(s)=\frac{1}{1-aG(s)} \end{equation} where $G(s)$ is the Laplace transform of a known probability density distribution $g(t)$ whose moments are $\langle t^n\rangle$.
Is it possible to approximate $f(t)$ in terms of the first moments $\langle t^n\rangle$? At least in the limit of large $t\gg1$?