The limit behaviour of random walks with arrests

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.