## Electronic Communications in Probability

### A quenched functional central limit theorem for planar random walks in random sceneries

#### Abstract

Random walks in random sceneries (RWRS) are simple examples of stochastic processes in disordered media. They were introduced at the end of the 70's by Kesten-Spitzer and Borodin, motivated by the construction of new self-similar processes with stationary increments. Two sources of randomness enter in their definition: a random field $\xi = (\xi(x))_{x \in \mathbb{Z}^d}$ of i.i.d. random variables, which is called the random scenery, and a random walk $S = (S_n)_{n \in \mathbb{N}}$ evolving in $\mathbb{Z}^d$, independent of the scenery. The RWRS $Z = (Z_n)_{n \in \mathbb{N}}$ is then defined as the accumulated scenery along the trajectory of the random walk, i.e., $Z_n := \sum_{k=1}^n \xi(S_k)$. The law of $Z$ under the joint law of $\xi$ and $S$ is called "annealed'', and the conditional law given $\xi$ is called "quenched''. Recently, functional central limit theorems under the quenched law were proved for $Z$ by the first two authors for a class of transient random walks including walks with finite variance in dimension $d \ge 3$. In this paper we extend their results to dimension $d=2$.

#### Article information

Source
Electron. Commun. Probab., Volume 19 (2014), paper no. 3, 9 pp.

Dates
Accepted: 28 January 2014
First available in Project Euclid: 7 June 2016

https://projecteuclid.org/euclid.ecp/1465316705

Digital Object Identifier
doi:10.1214/ECP.v19-3002

Mathematical Reviews number (MathSciNet)
MR3164750

Zentralblatt MATH identifier
1329.60075

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60G52: Stable processes

Rights

#### Citation

Guillotin-Plantard, Nadine; Poisat, Julien; Soares dos Santos, Renato. A quenched functional central limit theorem for planar random walks in random sceneries. Electron. Commun. Probab. 19 (2014), paper no. 3, 9 pp. doi:10.1214/ECP.v19-3002. https://projecteuclid.org/euclid.ecp/1465316705

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