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 $$ in a sparse random environment $$ is a nearest neighbor random walk on $$ that jumps to the left or to the right with probability 1/2 from every point of $$ and jumps to the right (left) with the random probability λ (1 - λ ) from the point S , $$ . Assuming that $$ are independent copies of a random vector $$ and the mean $$ is finite (moderate sparsity) we obtain stable limit laws for X , 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 $$ (strong sparsity) will be analyzed in a forthcoming paper.