Electronic Journal of Probability

Local asymptotics for the first intersection of two independent renewals

Kenneth S. Alexander and Quentin Berger

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K05: Renewal theory 60G50: Sums of independent random variables; random walks

intersection of renewal processes regular variations local asymptotics reverse renewal theorem coupling

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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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