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

Abstract

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∈N∪{0} in a sparse random environment (S k , λ k ) k∈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 λ k+1 (1 − λ k+1 ) from the point S k , k ∈ Z. Assuming that (S k − S k−1 , λ k ) k∈Z are independent copies of a random vector (ξ, λ) ∈ N × (0, 1) and the mean Eξ is finite (moderate sparsity) we obtain stable limit laws for X n , 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 Eξ = ∞ (strong sparsity) will be analyzed in a forthcoming paper.