Limit theorem for perturbed random walks

TL;DR

This paper proves that a class of perturbed random walks, which restart at zero and behave like different random walks on each side, converges to a skew Brownian motion under proper rescaling, linking renewal functions and transition probabilities.

Contribution

It establishes a limit theorem showing convergence of perturbed random walks to skew Brownian motion, incorporating renewal functions and transition probabilities.

Findings

Convergence of perturbed random walks to skew Brownian motion.

Explicit relation between model parameters and skew Brownian motion.

Framework for analyzing perturbed stochastic processes.

Abstract

We consider random walks perturbed at zero which behave like (possibly different) random walks with i.i.d. increments on each half lines and restarts at $0$ whenever they cross that point. We show that the perturbed random walk, after being rescaled in a proper way, converges to a skew Brownian motion whose parameter is defined by renewal functions of the simple random walks and the transition probabilities from $0$.