Electronic Journal of Probability

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

Elena Kosygina and Jonathon Peterson

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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.

Article information

Source
Electron. J. Probab., Volume 21 (2016), paper no. 70, 24 pp.

Dates
Received: 12 April 2016
Accepted: 11 November 2016
First available in Project Euclid: 2 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1480688087

Digital Object Identifier
doi:10.1214/16-EJP14

Mathematical Reviews number (MathSciNet)
MR3580036

Zentralblatt MATH identifier
1354.60030

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F17: Functional limit theorems; invariance principles 60J15

Keywords
excited random walk periodic cookie stacks Brownian motion perturbed at its extrema branching-like processes

Rights
Creative Commons Attribution 4.0 International License.

Citation

Kosygina, Elena; Peterson, Jonathon. Functional limit laws for recurrent excited random walks with periodic cookie stacks. Electron. J. Probab. 21 (2016), paper no. 70, 24 pp. doi:10.1214/16-EJP14. https://projecteuclid.org/euclid.ejp/1480688087


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References

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