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 , . Assuming that are independent copies of a random vector and the mean is finite (moderate sparsity) we obtain stable limit laws for , properly normalized and centered, as → . While the case ≤ a.s. for some deterministic > 0 (weak sparsity) was analyzed by Matzavinos et al., the case (strong sparsity) will be analyzed in a forthcoming paper.