The limit behaviour of random walks with arrests

O. O. Prykhodko
Theory of Stochastic Processes
Vol.24 (40), no.2, 2019, pp.79-88
Let 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 η_i≥0; and then continues its way as usual. The random variables η₁, η₂, … are assumed i.i.d. We study the limit behaviour of this process scaled as in the Donsker theorem. In case of E η_i < ∞, 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.