Limit theorem for perturbed random walks

Hoang-Long Ngo, Marc Peigné
Theory of Stochastic Processes
Vol.24 (40), no.2, 2019, pp.61-78
We consider random walks perturbed at zero which behave like (possibly different) random walk with independent and identically distributed 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 walk and the transition probabilities from 0.