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.
Cite
@article{arxiv.math/0611693,
title = {Nonlinear renewal theorems for random walks with perturbations of intermediate order},
author = {Keiji Nagai and Cun-Hui Zhang},
journal= {arXiv preprint arXiv:math/0611693},
year = {2007}
}
Comments
Published at http://dx.doi.org/10.1214/074921706000000671 in the IMS Lecture Notes--Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)