Random walks in a moderately sparse random environment

A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk ${\left(X_n\right)}_{n\in \mathbb{N}\cup \left\{0\right\}}$ in a sparse random environment ${(S_k,{\lambda }_k)}_{k\in \mathbb{Z}}$ is a nearest neighbor random walk on $\mathbb{Z}$ that jumps to the left or to the right with probability 1/2 from every point of $\mathbb{Z}\backslash \left\{\dots ,S_{-1},S_0=0,S_1,\dots \right\}$ and jumps to the right (left) with the random probability λ _k+1 (1 - λ _k+1) from the point S _k , $k\in \mathbb{Z}$ . Assuming that ${(S_k-S_{k-1},{\lambda }_k)}_{k\in \mathbb{Z}}$ are independent copies of a random vector $(\xi ,\lambda )\in \mathbb{N}\times (0,1)$ and the mean $E\xi$ is finite (moderate sparsity) we obtain stable limit laws for X _n , properly normalized and centered, as n → ∞. While the case ξ ≤ M a.s. for some deterministic M > 0 (weak sparsity) was analyzed by Matzavinos et al., the case $E\xi =\infty$ (strong sparsity) will be analyzed in a forthcoming paper.