• DOI:10.1214/ECP.V19-3002
• Corpus ID: 7420843

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

@article{GuillotinPlantard2013AQF,
  title={A quenched functional central limit theorem for planar random walks in random sceneries},
  author={Nadine Guillotin-Plantard and Julien Poisat and Renato Soares dos Santos},
  journal={Electronic Communications in Probability},
  year={2013},
  volume={19},
  pages={1-9}
}

• N. Guillotin-Plantard, J. Poisat, R. D. Santos
• Published 16 June 2013
• Mathematics
• Electronic Communications in Probability

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 $d=2$.