Asymptotic results for certain first-passage times and areas of renewal processes

We consider the process ${x-N(t):tgeq 0}$, where $x>0$ and ${N(t):tgeq 0}$ is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of $(tau(x),A(x))$ where $tau(x)$ is the first-passage time of ${x-N(t):tgeq 0}$ to reach zero or a negative value, and $A(x)$ is the corresponding first-passage area. We remark that we can define the sequence ${(tau(n),A(n)):ngeq 1}$ by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as $xtoinfty$ in the fashion of large (and moderate) deviations