Geodesic
Limit theorem for perturbed random walks
Teoriâ slučajnyh processov, Tome 24 (2019) no. 2, pp. 61-78.

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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 $0$.
Keywords: Invariance principle, Reflected Brownian motion, Renewal function, Skew Brownian motion. 
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Hoang-Long Ngo; Marc Peigné. Limit theorem for perturbed random walks. Teoriâ slučajnyh processov, Tome 24 (2019) no. 2, pp. 61-78. http://geodesic.mathdoc.fr/item/THSP_2019_24_2_a4/

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