Three-dimensional reflected driftless random walks in troughs: new asymptotic behavior

Abstract

The present work deals with reflected random walks in the trough Z + 2 × Z having a zero mean-drift in the interior of the domain. It is proved that in all non-critical cases the random walk is transient whereas it is recurrent in one critical subcase. The convergence problem for renormalized random walks is also studied. Two possible cases are separated. In the first one, so-called semimartingale case, the renormalized in a standard way random walks converge weakly to a semimartingale reflected Brownian motion described in [24], whereas in the second case the random walks exhibit a non-trivial behavior. Namely, one of its coordinates tends to infinity almost surely faster than √ n. The obtained results are based on the estimates of the invariant measure of a driftless random walk in a wedge derived in [6].