A Functional Central Limit Theorem for the General Brownian Motion on the Half-Line

TL;DR

This paper proves a new functional central limit theorem for Brownian motions on the half-line, using generator convergence and providing quantitative estimates, with applications to boundary-conditioned random walks.

Contribution

It introduces a Trotter-Kato type theorem for Feller processes and offers quantitative convergence estimates in the vague topology.

Findings

Established a Trotter-Kato type theorem for Feller processes.

Derived functional central limit theorems for boundary-conditioned random walks.

Provided quantitative estimates for process convergence.

Abstract

In this work, we establish a Trotter-Kato type theorem. More precisely, we characterize the convergence in distribution of Feller processes by examining the convergence of their generators. The main novelty lies in providing quantitative estimates in the vague topology at any fixed time. As important applications, we deduce functional central limit theorems for random walks on the positive integers with boundary conditions, which converge to Brownian motions on the positive half-line with boundary conditions at zero.