I know of only one instance where there is an explicit description of the error for the saddlepoint approximation of the cumulative distribution function (Janssen al. 2008, Adv. Appl. Prob., for Poisson CDF). This does not involve Fourier transform methods.
The paucity of theoretical results is in contrast to plentiful 'experimental' evidence (Butler, 2007, Cambridge UP) which shows that the saddlepoint approximation method tends to be excellent and not just in large IID samples.
As far as I can see the only hope for a universal error estimate is to go via bilateral Laplace transform (generalized Fourier) but I have no feel for how tight the resulting error bounds would be.
My question: is there a systematic method for computing tight error bounds for the saddlepoint CDF approximation? Has this been done?
