## Electronic Journal of Probability

### Large deviations and slowdown asymptotics for one-dimensional excited random walks

Jonathon Peterson

#### Abstract

We study the large deviations of excited random walks on $\mathbb{Z}$. We prove a large deviation principle for both the hitting times and the position of the random walk and give a qualitative description of the respective rate functions. When the excited random walk is transient with positive speed $v_0$, then the large deviation rate function for the position of the excited random walk is zero on the interval $[0,v_0]$ and so probabilities such as $P(X_n < nv)$ for $v \in (0,v_0)$ decay subexponentially. We show that rate of decay for such slowdown probabilities is polynomial of the order $n^{1-\delta/2}$, where $\delta>2$ is the expected total drift per site of the cookie environment.

#### Article information

Source
Electron. J. Probab., Volume 17 (2012), paper no. 48, 24 pp.

Dates
Accepted: 21 June 2012
First available in Project Euclid: 4 June 2016

https://projecteuclid.org/euclid.ejp/1465062370

Digital Object Identifier
doi:10.1214/EJP.v17-1726

Mathematical Reviews number (MathSciNet)
MR2946155

Zentralblatt MATH identifier
1260.60184

Rights

#### Citation

Peterson, Jonathon. Large deviations and slowdown asymptotics for one-dimensional excited random walks. Electron. J. Probab. 17 (2012), paper no. 48, 24 pp. doi:10.1214/EJP.v17-1726. https://projecteuclid.org/euclid.ejp/1465062370

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