Local asymptotics for the first intersection of two independent renewals

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.