I came across this nice paper where the author calculates the density of the sum of two iid Student's t variable using saddlepoint approximation, see (2.13).
I replicated this myself using the stationary point $\hat{y}=x/2$ but this clearly has a square-root singularity when $|x|=2\sqrt{\nu}$, so the approximation breaks down badly. However, the author claims it is a "quite good" approximation to the exact density and it is even plotted in the paper and looks indeed nice.
My question is: how do we do this properly? Do I have to take into account the other two stationary points? If yes, how? There is clearly a difficulty here as the three stationary points coalesce at $|x|=2\sqrt{\nu}$ and all become real.
