## The Annals of Probability

### A Renewal Theory with Varying Drift

Cun-Hui Zhang

#### 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

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