Local asymptotics for the first intersection of two independent renewals

Abstract

We study the intersection of two independent renewal processes, $\rho =\tau \cap \sigma$. Assuming that $\mathbf{P}({\tau }_1=n)=\phi (n)n^{-(1+\alpha )}$ and $\mathbf{P}({\sigma }_1=n)=\tilde{\phi }(n)n^{-(1+\tilde{\alpha })}$ for some $\alpha ,\tilde{\alpha }\ge 0$ and some slowly varying $\phi ,\tilde{\phi }$, 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.