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

Buraczewski, Dariusz||Dyszewski, Piotr||Iksanov, Alexander||Marynych, Alexander||Roitershtein, Alexander

Overview