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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.

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