= Saddlepoint density of a Laplace sample mean
{title2=$\widehat f_{\bar Y}(y)=\sqrt{\frac{n}{2\pi D(1+D)}}\exp\!\left[n\left\{\log\frac{1+D}{2}-D+1\right\}\right],\quad D=\sqrt{1+y^2}$}
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 $K'(t)=y$, giving $t=y/(1+\sqrt{1+y^2})$. At this tilt, $K''(t)=D(1+D)$ 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.
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