Local asymptotics for the first intersection of two independent renewals

TL;DR

This paper analyzes the asymptotic behavior of the intersection of two independent renewal processes, providing detailed probability estimates and insights into their first intersection times, with applications to coupling bounds.

Contribution

It offers new asymptotic results for the intersection renewal process, including the first intersection probability, extending renewal theory in a reverse manner.

Findings

Derived asymptotics for ( ho_1 > n) and ( ho_1 = n)

Identified behavior in the critical case where ( au_1) = 1

Provided bounds for renewal mass function increments

Abstract

We study the intersection of two independent renewal processes, $\rho=\tau\cap\sigma$. Assuming that $\mathbf{P}(\tau_1 = n ) = \varphi(n)\, n^{-(1+\alpha)}$ and $\mathbf{P}(\sigma_1 = n ) = \tilde\varphi(n)\, n^{-(1+ \tilde\alpha)} $ for some $\alpha,\tilde \alpha \geq 0$ and some slowly varying $\varphi,\tilde\varphi$, we give the asymptotic behavior first of $\mathbf{P}(\rho_1>n)$ (which is straightforward except in the case of $\min(\alpha,\tilde\alpha)=1$) and then of $\mathbf{P}(\rho_1=n)$. The result may be viewed as a kind of reverse renewal theorem, as we determine probabilities $\mathbf{P}(\rho_1=n)$ while knowing asymptotically the renewal mass function $\mathbf{P}(n\in\rho)=\mathbf{P}(n\in\tau)\mathbf{P}(n\in\sigma)$. Our results can be used to bound coupling-related quantities, specifically the increments $|\mathbf{P}(n\in\tau)-\mathbf{P}(n-1\in\tau)|$ of the renewal mass function.