Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 32 / 3 / ii / Solution

For the unit-scale Laplace distribution, integrating separately over the two half-lines gives the moment-generating functionfor $|t|<1$; the appropriate half-line integral diverges outside that interval. ThereforeIts derivatives needed for the saddlepoint density approximation areSet $D=\sqrt{1+y^2}$. Solving $K^{\prime }(\hat{t})=y$ and selecting the root in $(-1,1)$ gives the stable formulaincluding $\hat{t}=0$ at $y=0$. The other quadratic root is outside the domain. At the correct saddlepoint,Substitution gives the saddlepoint density of a Laplace sample mean:It is symmetric in $y$, as the underlying Laplace distribution requires. At zero its value is $\sqrt{n/(4\pi )}$, consistent with the local normal approximation having variance $2/n$. This is an approximation, not an exact finite-$n$ density or an automatically normalized one.