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

A. Yu. Pilipenko, Yu. E. Prykhodko
Theory of Stochastic Processes
Vol.19 (35), no.1, 2014, pp.52-61
Consider a Markov chain on $\Z_+$ 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.