Inverse Gaussian sum closure ($S_n\sim IG(\varphi ,n^2\lambda )$)

In the parametrization with density proportional to $y^{-3/2}\exp [-(\lambda /y+\varphi y)/2]$, the Inverse Gaussian distribution has cumulant-generating function $K(t)=\sqrt{\lambda \varphi }-\sqrt{\lambda (\varphi -2t)}$. For $n$ independent copies, the average has this same family with parameters $(n\varphi ,n\lambda )$, while the sum has parameters $(\varphi ,n^2\lambda )$. These assertions follow by multiplying moment-generating functions and scaling the density. The more usual mean parameter is $\mu =\sqrt{\lambda /\varphi }$, so the sum has mean $n\mu$ and shape $n^2\lambda$.

For the inverse Gaussian sum closure, take $x=s/n>0$. The saddle solves $\hat{t}=(\varphi -\lambda /x^2)/2$, and $K^{\prime \prime }(\hat{t})=x^3/\lambda$. Substitution in the saddlepoint density approximation givesThis is the exact sum density, so its leading saddlepoint expression needs no correction or renormalization.