Functional limit laws for recurrent excited random walks with periodic cookie stacks

Abstract

We consider one-dimensional excited random walks (ERWs) with periodic cookie stacks in the recurrent regime. We prove functional limit theorems for these walks which extend the previous results in [DK12] for excited random walks with “boundedly many cookies per site.” In particular, in the non-boundary recurrent case the rescaled excited random walk converges in the standard Skorokhod topology to a Brownian motion perturbed at its extrema (BMPE). While BMPE is a natural limiting object for excited random walks with boundedly many cookies per site, it is far from obvious why the same should be true for our model which allows for infinitely many “cookies” at each site. Moreover, a BMPE has two parameters $\alpha ,\beta <1$ and the scaling limits in this paper cover a larger variety of choices for $\alpha $ and $\beta $ than can be obtained for ERWs with boundedly many cookies per site.