Extend Markov nonlinear renewal theory to multivariate settings

Extend the nonlinear Markov renewal theory for one-dimensional Markov random walks with slowly changing perturbations and nonlinear boundaries to multivariate settings. Specifically, establish asymptotic expansions and limit theorems for boundary-crossing times, overshoot distributions, and expected stopping times for multidimensional Markov random walks when the crossing boundary depends nonlinearly on time and the multivariate additive components, thereby generalizing the existing univariate nonlinear Markov renewal results to the multivariate case.

Background

This article reviews classical nonlinear renewal theory for i.i.d. processes and its Markov extension to one-dimensional settings, where boundary-crossing times and overshoot distributions are analyzed for perturbed random walks crossing nonlinear boundaries. It then surveys multivariate Markov renewal theory—largely for linear boundaries—and recent multivariate nonlinear renewal results in the i.i.d. case, highlighting the breadth of applications in sequential analysis and change detection.

Despite these advances, a comprehensive framework for multivariate nonlinear Markov renewal theory, which would unify multivariate dependence with nonlinear boundary behavior in Markov random walks, has not yet been developed. Such a theory would parallel the univariate nonlinear Markov renewal results by establishing asymptotic distributions and moment expansions for first-passage events in multidimensional Markov-dependent processes.