## Electronic Journal of Probability

### Local asymptotics for the first intersection of two independent renewals

#### 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.

#### Article information

Source
Electron. J. Probab. Volume 21, Number (2016), paper no. 68, 20 pp.

Dates
Accepted: 25 November 2016
First available in Project Euclid: 1 December 2016

https://projecteuclid.org/euclid.ejp/1480561217

Digital Object Identifier
doi:10.1214/16-EJP17

#### Citation

Alexander, Kenneth S.; Berger, Quentin. Local asymptotics for the first intersection of two independent renewals. Electron. J. Probab. 21 (2016), paper no. 68, 20 pp. doi:10.1214/16-EJP17. https://projecteuclid.org/euclid.ejp/1480561217

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