## Electronic Communications in Probability

- Electron. Commun. Probab.
- Volume 22 (2017), paper no. 39, 9 pp.

### A functional limit theorem for excited random walks

#### Abstract

We consider the limit behavior of an excited random walk (ERW), i.e., a random walk whose transition probabilities depend on the number of times the walk has visited to the current state. We prove that an ERW being naturally scaled converges in distribution to an excited Brownian motion that satisfies an SDE, where the drift of the unknown process depends on its local time. Similar result was obtained by Raimond and Schapira, their proof was based on the Ray-Knight type theorems. We propose a new method based on a study of the Radon-Nikodym density of the ERW distribution with respect to the distribution of a symmetric random walk.

#### Article information

**Source**

Electron. Commun. Probab., Volume 22 (2017), paper no. 39, 9 pp.

**Dates**

Received: 11 November 2016

Accepted: 14 June 2017

First available in Project Euclid: 9 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ecp/1502244193

**Digital Object Identifier**

doi:10.1214/17-ECP66

**Mathematical Reviews number (MathSciNet)**

MR3685237

**Zentralblatt MATH identifier**

06797792

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments 60F17: Functional limit theorems; invariance principles

**Keywords**

excited random walks excited Brownian motion invariance principle

**Rights**

Creative Commons Attribution 4.0 International License.

#### Citation

Pilipenko, Andrey. A functional limit theorem for excited random walks. Electron. Commun. Probab. 22 (2017), paper no. 39, 9 pp. doi:10.1214/17-ECP66. https://projecteuclid.org/euclid.ecp/1502244193