A high order expansion of the renewal function is provided under the assumption that the inter-renewal time distribution is light tailed with finite moment generating function $g$ on a neighborhood of $0$. This expansion relies on complex analysis and is expressed in terms of the residues of the function $1/(1-g)$. Under the assumption that $g$ can be extended into a meromorphic function on the complex plane and some technical conditions, we obtain even an exact expansion of the renewal function. An application to risk theory is given where we consider high order expansion of the ruin probability for the standard compound Poisson risk model. This precises the well- known Crámer-Lundberg approximation of the ruin probability when the initial reserve is large.
"High order expansions for renewal functions and applications to ruin theory." Ann. Appl. Probab. 27 (4) 2342 - 2382, August 2017. https://doi.org/10.1214/16-AAP1261