Proofs of the martingale FCLT

TL;DR

This paper provides an in-depth review of the proof techniques for the martingale functional central limit theorem, emphasizing tightness criteria and the martingale method in stochastic-process limits for queueing models.

Contribution

It offers a detailed exposition of the proof of the martingale FCLT and discusses tightness criteria, enhancing understanding of stochastic-process limits in queueing theory.

Findings

Clarifies the proof of the martingale FCLT.

Highlights one-dimensional tightness criteria.

Supports diffusion-process approximations in queueing models.

Abstract

This is an expository review paper elaborating on the proof of the martingale functional central limit theorem (FCLT). This paper also reviews tightness and stochastic boundedness, highlighting one-dimensional criteria for tightness used in the proof of the martingale FCLT. This paper supplements the expository review paper Pang, Talreja and Whitt (2007) illustrating the ``martingale method'' for proving many-server heavy-traffic stochastic-process limits for queueing models, supporting diffusion-process approximations.