Limit behavior of a simple random walk with non-integrable jump from a barrier

Consider a Markov chain on $\mathbb{Z}_+$ with reflecting barrier at 0 such that jumps of the chain outside of 0 are i.i.d. with mean zero and finite variance. It is assumed that the jump from 0 has a distribution that belongs to the domain of attraction of non-negative stable law. It is proved that under natural scaling of a space and a time a limit of this scaled Markov chain is a Brownian motion with some Wentzell's boundary condition at 0.