A note on the asymptotics of lattice paths with general boundaries

Lattice paths which are restricted by one or two nonlinear boundaries are of increasing importance in many applications of lattice path combinatorics, for instance in sequential statistics. It is typical for such applications that the number of paths which avoid certain boundaries has to be determined when the number of steps is large. Counting results, if available, and recursion are not very well suited in these cases. In this paper we present some asymptotic approximations which are easy to calculate and provide an accuracy which is sufficient for many practical purposes.