## The Annals of Probability

- Ann. Probab.
- Volume 16, Number 2 (1988), 793-824.

### A Nonlinear Renewal Theory

#### Abstract

Let $T$ be the first time that a perturbed random walk crosses a nonlinear boundary. This paper concerns the approximations of the distribution of the excess over the boundary, the expected stopping time $ET$ and the variance of the stopping time $\operatorname{Var}(T)$. Expansions are obtained by using linear renewal theorems with varying drift.

#### Article information

**Source**

Ann. Probab., Volume 16, Number 2 (1988), 793-824.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176991788

**Digital Object Identifier**

doi:10.1214/aop/1176991788

**Mathematical Reviews number (MathSciNet)**

MR929079

**Zentralblatt MATH identifier**

0643.60067

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K05: Renewal theory

Secondary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60J15 62L10: Sequential analysis 62L12: Sequential estimation 62L15: Optimal stopping [See also 60G40, 91A60]

**Keywords**

Nonlinear renewal theory excess over the boundary uniform integrability expected sample size variance of sample size

#### Citation

Zhang, Cun-Hui. A Nonlinear Renewal Theory. Ann. Probab. 16 (1988), no. 2, 793--824. doi:10.1214/aop/1176991788. https://projecteuclid.org/euclid.aop/1176991788