Limiting saddlepoint relative errors in large deviation regions under purely Tauberian conditions

Most theoretical results on the relative errors of saddlepoint approximations in the extreme tails have involved placing conditions on the density/mass function. Checking the validity of such conditions is problematic when density/mass functions are intractable, as is typically the case in important practical applications involving convolved, compound, and first-passage distributions as well as for moment generating functions MGFs that are regularly varying. In this paper, we present novel conditions which ensure the existence of positive finite limiting relative errors for saddlepoint density/mass function and survival function approximations. These conditions, which are rather weak, are expressed entirely in terms of the MGF, hence the description purely Tauberian. We focus mainly on the cases in which there are positive and negative gamma distributional limits (the only other non-degenerate possibility being a Gaussian limit) and we show how to check the new conditions in important classes of models in these two settings.