Some Remarks on the Large Deviation of the Visited Sites of Simple Random Walk in Random Scenery

Parkpoom Phetpradap

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  • Support Team

Keywords:

random walk, random scenery, large deviation principle, moderate deviation principle, sums of independent random variables

Abstract

For each $z \in \mathbb{Z}^d$, we define random scenery on the integer lattice $\mathbb{Z}^d$ as $\{ \xi_z: z \in \mathbb{Z}^d\}$ where each $\xi_z$ are identical and independent random variables with finite mean and variance. For a simple symmetric random walk on $\mathbb{Z}^d$ in dimension $d \geq 3$, we focus on $X_n:= \sum_{z \in V_n} \xi_z $, where $V_n$ is the lattice visited by the walk by time $n$. We investigate that $X_n$ satisfies large deviation principle with explicitly given rate functions. The expectation and variance of $X_n$ can also be calculated. This is an extended result from the large deviation result on the number of sites visited by a simple random walk.

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Published

2023-05-16

How to Cite

Team, S. (2023). Some Remarks on the Large Deviation of the Visited Sites of Simple Random Walk in Random Scenery: Parkpoom Phetpradap. Thai Journal of Mathematics, 217–226. Retrieved from https://thaijmath2.in.cmu.ac.th/index.php/thaijmath/article/view/770