Asymptotic results for certain first-passage times and areas of renewal processes (2105.07978v3)

Abstract: We consider the process ${x-N(t):t\geq 0}$, where $x\in\mathbb{R}_+$ and ${N(t):t\geq 0}$ 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 ${x-N(t):t\geq 0}$ to reach zero or a negative value, and $A(x):=\int_0^{\tau(x)}(x-N(t))dt$ is the corresponding first-passage (positive) area swept out by the process ${x-N(t):t\geq 0}$. We remark that we can define the sequence ${(\tau(n),A(n)):n\geq 1}$ by referring to the concept of integrated random walk. Our aim is to prove asymptotic results as $x\to\infty$ in the fashion of large (and moderate) deviations.