The weak convergence of regenerative processes using some excursion path decompositions

TL;DR

This paper establishes general conditions for the weak convergence of regenerative processes in Polish spaces by analyzing the convergence of significant excursions and their endpoints, with applications to processes built from concatenated i.i.d. paths.

Contribution

It introduces a unified framework for proving weak convergence of regenerative processes via excursion decompositions and provides sufficient conditions on excursion measures.

Findings

Established a general convergence criterion involving psilon-big excursions.

Provided sufficient conditions on excursion measures for convergence.

Discussed extensions to concatenated i.i.d. path processes.

Abstract

We consider regenerative processes with values in some Polish space. We define their \epsilon-big excursions as excursions e such that f(e)>\epsilon, where f is some given functional on the space of excursions which can be thought of as, e.g., the length or the height of e. We establish a general condition that guarantees the convergence of a sequence of regenerative processes involving the convergence of \epsilon-big excursions and of their endpoints, for all \epsilon in a countable set whose closure contains 0. Finally, we provide various sufficient conditions on the excursion measures of this sequence for this general condition to hold and discuss possible generalizations of our approach to processes that can be written as the concatenation of i.i.d. paths.