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Local asymptotics for the first intersection of two independent renewals

We study the intersection of two independent renewal processes, $ρ=τ\capσ$. Assuming that $\mathbf{P}(τ_1 = n ) = φ(n)\, n^{-(1+α)}$ and $\mathbf{P}(σ_1 = n ) = \tildeφ(n)\, n^{-(1+ \tildeα)} $ for some $α,\tilde α\geq 0$ and some slowly varying $φ,\tildeφ$, we give the asymptotic behavior first of $\mathbf{P}(ρ_1>n)$ (which is straightforward except in the case of $\min(α,\tildeα)=1$) and then of $\mathbf{P}(ρ_1=n)$. The result may be viewed as a kind of reverse renewal theorem, as we determine probabilities $\mathbf{P}(ρ_1=n)$ while knowing asymptotically the renewal mass function $\mathbf{P}(n\inρ)=\mathbf{P}(n\inτ)\mathbf{P}(n\inσ)$. Our results can be used to bound coupling-related quantities, specifically the increments $|\mathbf{P}(n\inτ)-\mathbf{P}(n-1\inτ)|$ of the renewal mass function.