## Institute of Mathematical Statistics Lecture Notes - Monograph Series

- Lecture Notes--Monograph Series
- Number 50, 2006, 164-175

### Nonlinear renewal theorems for random walks with perturbations of intermediate order

#### Abstract

We develop nonlinear renewal theorems for a perturbed random walk without assuming stochastic boundedness of centered perturbation terms. A second order expansion of the expected stopping time is obtained via the uniform integrability of the difference between certain linear and nonlinear stopping rules. An intermediate renewal theorem is obtained which provides expansions between the nonlinear versions of the elementary and regular renewal theorems. The expected sample size of a two-sample rank sequential probability ratio test is considered as the motivating example.

#### Chapter information

**Source***Recent Developments in Nonparametric Inference and Probability: Festschrift for Michael Woodroofe* (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2006)

**Dates**

First available in Project Euclid: 28 November 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.lnms/1196284060

**Digital Object Identifier**

doi:10.1214/074921706000000671

**Mathematical Reviews number (MathSciNet)**

MR2409551

**Zentralblatt MATH identifier**

1268.60109

**Subjects**

Primary: 60K05: Renewal theory 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 62L10: Sequential analysis

**Keywords**

nonlinear renewal theorem random walk sequential analysis expected stopping rule uniform integrability sequential probability ratio test rank likelihood rank test proportional hazards model

**Rights**

Copyright © 2006, Institute of Mathematical Statistics

#### Citation

Nagai, Keiji; Zhang, Cun-Hui. Nonlinear renewal theorems for random walks with perturbations of intermediate order. Recent Developments in Nonparametric Inference and Probability, 164--175, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2006. doi:10.1214/074921706000000671. https://projecteuclid.org/euclid.lnms/1196284060