Geodesic
The limit behaviour of random walks with arrests
Teoriâ slučajnyh processov, Tome 24 (2019) no. 2, pp. 79-88.

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Let $\tilde S$ be a random walk which behaves like a standard centred and square-integrable random walk except when hitting $0$. Upon the $i$-th hit of $0$ the random walk is arrested there for a random amount of time $\eta_i \geq 0$; and then continues its way as usual. The random variables $\eta_1, \ \eta_2, \ \ldots$ are assumed i.i.d. We study the limit behaviour of this process scaled as in the Donsker theorem. In case of $\mathbb E \eta_i \infty$, weak convergence towards a Wiener process is proved. We also consider the sequence of processes whose arrest times are geometrically distributed and grow with $n$. We prove that the weak limit for the last model is either a Wiener process, a Wiener process stopped at 0 or a Wiener process with a sticky point.
Keywords: Functional limit theorem, sticky Brownian motion, perturbed random walks. 
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O. O. Prykhodko. The limit behaviour of random walks with arrests. Teoriâ slučajnyh processov, Tome 24 (2019) no. 2, pp. 79-88. http://geodesic.mathdoc.fr/item/THSP_2019_24_2_a5/

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