Geodesic
Limit behavior of a simple random walk with non-integrable jump from a barrier
Teoriâ slučajnyh processov, Tome 19 (2014) no. 1, pp. 52-61

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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.
Keywords: Random walk; Wentzell's boundary condition; invariance principle. 
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     author = {A. Yu. Pilipenko and Yu. E. Prykhodko},
     title = {Limit behavior of a simple random walk with non-integrable jump from a barrier},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {52--61},
     publisher = {mathdoc},
     volume = {19},
     number = {1},
     year = {2014},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2014_19_1_a5/}
}
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A. Yu. Pilipenko; Yu. E. Prykhodko. Limit behavior of a simple random walk with non-integrable jump from a barrier. Teoriâ slučajnyh processov, Tome 19 (2014) no. 1, pp. 52-61. http://geodesic.mathdoc.fr/item/THSP_2014_19_1_a5/