On Tightness of the Skew Random Walks

TL;DR

This paper proves the tightness of skew random walks, demonstrating that skew Brownian motion can be obtained as their scaling limit, using a fourth-order moment method for the proof.

Contribution

It introduces a new tightness proof for skew random walks, enabling their use as approximations for skew Brownian motion.

Findings

Skew random walks are tight under scaling.

Skew Brownian motion can be constructed as a limit of skew random walks.

A fourth-order moment method is effective for proving tightness.

Abstract

The primary purpose of this article is to prove a tightness of skew random walks. The tightness result implies, in particular, that the skew Brownian motion can be constructed as the scaling limit of such random walks. Our proof of tightness is based on a fourth-order moment method.