Saddlepoint density of a Laplace sample mean (${\hat{f}}_{\bar{Y}}(y)=\sqrt{\frac{n}{2\pi D(1+D)}}\exp \left[n\left\{\log \frac{1+D}{2}-D+1\right\}\right],D=\sqrt{1+y^2}$)

For independent unit-scale Laplace distributions, the cumulant-generating function is $K(t)=-\log (1-t^2)$ for $|t|<1$. The saddlepoint density approximation solves $K^{\prime }(t)=y$, giving $t=y/(1+\sqrt{1+y^2})$. At this tilt, $K^{\prime \prime }(t)=D(1+D)$ and $ty=D-1$, giving the displayed probability density function approximation for the sample mean. Its value at zero is $\sqrt{n/(4\pi )}$; it need not integrate to one before normalization.