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VOL. 50 | 2006 Nonlinear renewal theorems for random walks with perturbations of intermediate order
Keiji Nagai, Cun-Hui Zhang

Editor(s) Jiayang Sun, Anirban DasGupta, Vince Melfi, Connie Page


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.


Published: 1 January 2006
First available in Project Euclid: 28 November 2007

zbMATH: 1268.60109
MathSciNet: MR2409551

Digital Object Identifier: 10.1214/074921706000000671

Primary: 60G40 , 60K05 , 60K35
Secondary: 62L10

Keywords: expected stopping rule , nonlinear renewal theorem , proportional hazards model , Random walk , rank likelihood , rank test , sequential analysis , sequential probability ratio test , uniform integrability

Rights: Copyright © 2006, Institute of Mathematical Statistics


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