The Annals of Probability

A Renewal Theory with Varying Drift

Cun-Hui Zhang

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Abstract

Let $R$ be the excess over the boundary in renewal theory. It is well known that $ER$ has a limit $r$ when the drift of the random walk $\mu \geq 0$. We study renewal theorems with varying $\mu$. Conditions are given under which the tail $ER - r$ is uniformly dominated by a decreasing integrable function for $\mu$ in a compact interval in $(0, \infty)$. Conditions are also given under which the derivative of the tail $(\partial/\partial\mu)(ER - r)$ is uniformly dominated by a directly Riemann integrable function.

Article information

Source
Ann. Probab., Volume 17, Number 2 (1989), 723-736.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176991423

Digital Object Identifier
doi:10.1214/aop/1176991423

Mathematical Reviews number (MathSciNet)
MR985386

Zentralblatt MATH identifier
0676.60080

JSTOR
links.jstor.org

Subjects
Primary: 60K05: Renewal theory
Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60]

Keywords
Renewal theory uniform convergence excess over the boundary Fourier transformation

Citation

Zhang, Cun-Hui. A Renewal Theory with Varying Drift. Ann. Probab. 17 (1989), no. 2, 723--736. doi:10.1214/aop/1176991423. https://projecteuclid.org/euclid.aop/1176991423


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