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Local asymptotics for the first intersection of two independent renewals. (English) Zbl 1354.60108
Summary: 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 $$\alpha ,\tilde{\alpha } \geq 0$$ 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.

##### MSC:
 60K05 Renewal theory 60F99 Limit theorems in probability theory 60F10 Large deviations 60G50 Sums of independent random variables; random walks
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