Quenched Central Limit Theorems for Random Walks in Random Scenery

TL;DR

This paper proves quenched central limit theorems for random walks in random scenery, showing that under certain conditions, the normalized sums converge almost surely to a Gaussian distribution with explicit variance.

Contribution

It establishes quenched CLTs for random walks in random scenery with explicit variance formulas, advancing understanding of their asymptotic behavior.

Findings

Almost sure convergence to Gaussian law

Explicit variance formulas derived

Conditions on walk and scenery specified

Abstract

Random walks in random scenery are processes defined by $$Z_n:=\sum_{k=1}^n\omega_{S_k}$$ where $S:=(S_k,k\ge 0)$ is a random walk evolving in $\mathbb{Z}^d$ and $\omega:=(\omega_x, x\in{\mathbb Z}^d)$ is a sequence of i.i.d. real random variables. Under suitable assumptions on the random walk $S$ and the random scenery $\omega$, almost surely with respect to $\omega$, the correctly renormalized sequence $(Z_n)_{n\geq 1}$ is proved to converge in distribution to a centered Gaussian law with explicit variance.