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 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 k+1 (1 - λ k+1) from the point S k , . Assuming that are independent copies of a random vector and the mean 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 (strong sparsity) will be analyzed in a forthcoming paper.
Highlights
In the present article we investigate an intermediate model, called random walk in a sparse random environment (RWSRE), in which homogeneity of an environment is only perturbed on a sparse subset of Z
One assumes that an environment ω forms a stationary and ergodic sequence or even a sequence of iid random variables. In this setting random walk in a random environment (RWRE) has attracted a fair amount of attention among probabilistic community resulting in quenched and annealed limit theorems [3, 11, 12, 25, 26, 35, 37] and large deviations [5, 7, 9, 15, 19, 33, 34, 38, 39]
This shows, among others, that the Markov chain (Rk)k∈N0 is an instance of the random difference equation which corresponds to (A, B) = (ρ, ρξ)
Summary
Simple random walks on Z (the set of integers) arise in various areas of classical and modern stochastics. One assumes that an environment ω forms a stationary and ergodic sequence or even a sequence of iid (independent and identically distributed) random variables In this setting RWRE has attracted a fair amount of attention among probabilistic community resulting in quenched and annealed limit theorems [3, 11, 12, 25, 26, 35, 37] and large deviations [5, 7, 9, 15, 19, 33, 34, 38, 39].
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