Limit theorem for perturbed random walks

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 $0$.