Stable limit laws for random walk in a sparse random environment I: moderate sparsity

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 $(X_n)_{nin mathbb{N}cup{0}}$ in a sparse random environment $(S_k,lambda_k)_{kinmathbb{Z}}$ 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 $mathbb{Z}setminus {ldots,S_{-1},S_0=0,S_1,ldots}$ and jumps to the right (left) with the random probability $lambda_{k+1}$ ($1-lambda_{k+1}$) from the point $S_k$, $kinmathbb{Z}$. Assuming that $(S_k-S_{k-1},lambda_k)_{kinmathbb{Z}}$ are independent copies of a random vector $(xi,lambda)in mathbb{N}times (0,1)$ and the mean $mathbb{E}xi$ is finite (moderate sparsity) we obtain stable limit laws for $X_n$, properly normalized and centered, as $ntoinfty$. While the case $xileq M$ a.s. for some deterministic $M>0$ (weak sparsity) was analyzed by Matzavinos et al., the case $mathbb{E} xi=infty$ (strong sparsity) will be analyzed in a forthcoming paper.