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 (X-n)(n is an element of N)boolean OR{o} in a sparse random environment (S-k, lambda(k)) k is an element of z is a nearest neighbor random walk on Z that jumps to the left or to the right with probability 1/2 from every point of Z \ {... , S-1, S-0 = 0, S-1, ...} and jumps to the right (left) with the random probability lambda(k+1) (1 - lambda(k+1) ) from the point S-k, k is an element of Z. Assuming that (S-k - Sk-1, lambda(k)) k is an element of z are independent copies of a random vector (xi, lambda) is an element of IN x (0,1) and the mean IE is finite (moderate sparsity) we obtain stable limit laws for X-n, properly normalized and centered, as n -> infinity. While the case xi <= M a.s. for some deterministic M > 0 (weak sparsity) was analyzed by Matzavinos et al., the case E xi = infinity (strong sparsity) will be analyzed in a forthcoming paper.