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

TL;DR

This paper derives asymptotic behaviors for the joint distribution of first-passage times and areas in renewal processes with light-tailed distributions, extending understanding of large deviations in such stochastic models.

Contribution

It provides new asymptotic results for the joint distribution of first-passage times and areas in renewal processes, linking to integrated random walks.

Findings

Asymptotic formulas for large x

Results applicable to light-tailed renewal processes

Extension to integrated random walk concepts

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.