 Lugannani-Rice saddlepoint tail approximation

For a sample mean from an independent identically distributed sample, let $K(t)$ be the cumulant-generating function of one observation and solve $K^{\prime }(t)=x$. Put $w=\mathrm{sgn}(t)\sqrt{2n(tx-K(t))}$ and $u=t\sqrt{n{K}^{\prime \prime }(t)}$. Under the smooth, nonlattice conditions for a saddlepoint density approximation, the corresponding approximation to the cumulative distribution function iswhere $\Phi$ and $\varphi$ are the standard normal distribution functions. The apparent singularity at $t=0$ is removable. Write $v=K^{\prime \prime }(0)$ and $k=K^{\prime \prime \prime }(0)$; Taylor expansion gives $w=t\sqrt{nv}(1+kt/(3v)+O(t^2))$ and $u=t\sqrt{nv}(1+kt/(2v)+O(t^2))$. Thus $1/w-1/u\to k/(6\sqrt{n}{v}^{3/2})$. The approximation requires suitable tail regularity and is not a universal formula for distributions without a moment-generating function.