Tail bounds for the distribution of the deficit in the renewal risk model

We obtain upper and lower bounds for the tail of the deficit at ruin in the renewal risk model, which are (i) applicable generally; and (ii) based on reliability classifications. We also derive two-side bounds, in the general case where a function satisfies a defective renewal equation, and we apply them to the renewal model, using the function [Lambda]u introduced by [Psarrakos, G., Politis, K., 2007. A generalisation of the Lundberg condition in the Sparre Andersen model and some applications (submitted for publication)]. Finally, we construct an upper bound for the integrated function and an asymptotic result when the adjustment coefficient exists.