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

We consider the process {x − N(t) : t ≥ 0}, where x ∈ R+ and {N(t) :t ≥ 0} is a renewal process with light-tailed distributed holding times. We are interested in the joint distribution of (τ(x),A(x)) where τ(x) is the first-passage time of {x−N(t) : t ≥ 0} to reach zero or a negative value, and A(x) := \int_0^{τ(x)} (x−N(t))dt is the corresponding first-passage (positive) area swept out by the process {x−N(t) :t ≥ 0}. We remark that we can define the sequence {(τ(n),A(n)) : n ≥ 1} by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as x→∞ in the fashion of large (and moderate) deviations.