= Exact saddlepoint density of an inverse Gaussian sum
{title2=$\widehat f_{S_n}(s)=f_{S_n}(s)$}
For the <inverse Gaussian sum closure>, take $x=s/n>0$. The saddle solves $\widehat t=(\phi-\lambda/x^2)/2$, and $K''(\widehat t)=x^3/\lambda$. Substitution in the <saddlepoint density approximation> gives
$$
\widehat f_{S_n}(s)=\frac{n\sqrt\lambda}{\sqrt{2\pi}s^{3/2}}\exp\left[n\sqrt{\lambda\phi}-\frac12\left(\frac{n^2\lambda}{s}+\phi s\right)\right].
$$
This is the exact sum density, so its leading saddlepoint expression needs no correction or renormalization.
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