Functional limit laws for recurrent excited random walks with periodic cookie stacks. (English) Zbl 1354.60030

Kosygina, Elena; Peterson, Jonathon

Electron. J. Probab. 21, Paper No. 70, 24 p. (2016).

Summary: 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 [D. Dolgopyat and the first author, Electron. Commun. Probab. 17, Paper No. 35, 14 p. (2012; Zbl 1252.60098)] 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.

MSC:

60F17	Functional limit theorems; invariance principles
60G50	Sums of independent random variables; random walks
60K35	Interacting random processes; statistical mechanics type models; percolation theory
60J65	Brownian motion
60J80	Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

excited random walk; functional limit laws; periodic cookie stacks; Brownian motion; perturbation; extrema; branching-like processes

Citations:

Zbl 1252.60098