Residue expansions and saddlepoint approximations in stochastic models using the analytic continuation of generating functions

Asymptotic residue expansions are proposed for inverting probability generating functions (PGFs) and approximating their associated mass survival functions. The useful in the wide range of stochastic model applications which a PGF admits poles its analytic continuation. error such an expansion is contour integral continuation saddlepoint approximations developed errors using method steepest descents. These estimates attain sufficient accuracy that they can be used to set order so it achieves specified error. Numerical include success run tutorial example, discrete ruin model, Pollaczek-Khintchine formula, passage times semi-Markov processes. apply more generally arise renewal theory combinatorics lead simple proof classic theorem. They extend even further determining Taylor coefficients general meromorphic