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2016 Local asymptotics for the first intersection of two independent renewals
Kenneth S. Alexander, Quentin Berger
Electron. J. Probab. 21: 1-20 (2016). DOI: 10.1214/16-EJP17

Abstract

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.

Citation

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Kenneth S. Alexander. Quentin Berger. "Local asymptotics for the first intersection of two independent renewals." Electron. J. Probab. 21 1 - 20, 2016. https://doi.org/10.1214/16-EJP17

Information

Received: 23 March 2016; Accepted: 25 November 2016; Published: 2016
First available in Project Euclid: 1 December 2016

zbMATH: 1354.60108
MathSciNet: MR3580034
Digital Object Identifier: 10.1214/16-EJP17

Subjects:
Primary: 60G50, 60K05

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