Convergence of excursion point processes and its applications to functional limit theorems of Markov processes on a half-line

TL;DR

This paper establishes invariance principles for Markov processes on a half-line by analyzing the convergence of excursion point processes, leveraging Itô's excursion theory and recent advances in excursion measure convergence.

Contribution

It introduces a novel approach to proving invariance principles for Markov processes using excursion point process convergence and extends understanding of domains of attraction for self-similar processes.

Findings

Proves invariance principles for Markov processes on a half-line.

Characterizes domains of attraction for different self-similar processes.

Utilizes advanced excursion measure convergence techniques.

Abstract

Invariance principles are obtained for a Markov process on a half-line with continuous paths on the interior. The domains of attraction of the two different types of self-similar processes are investigated. Our approach is to establish convergence of excursion point processes, which is based on It\^{o}'s excursion theory and a recent result on convergence of excursion measures by Fitzsimmons and the present author.