Strongly correlated random fields as observed by a random walker

Abstract

We are given a random walk S 1, S 2, ... on ℤν, ν≧1, and a strongly correlated stationary random field ξ(x), xeℤν, which is independent of the random walk. We consider the field as observed by a random walker and study partial sums of the form \(W_n = \sum\limits_{j = {\text{ }}1}^n {\xi (S_j )}\). It is assumed that the law corresponding to the random walk belongs to the domain of attraction of a non-degenerate stable law of index β, 0<β≦2. We further suppose that the field ξ satisfies the non-central limit theorem of Dobrushin and Major with a scaling factor \(n^{ - v + \tfrac{1}{2}\alpha k} ,\alpha k < v\). Under the assumption αk<β it is shown that \(n^{ - 1 + \tfrac{1}{2}\alpha k/\beta } {\text{ }}W_{[nt]} \) converges weakly as n→∞ to a self-similar process {Δ t , t≧0} with stationary increments, and Δ t can be represented as a multiple Wiener-Ito integral of a random function. This extends the noncentral limit theorem of Dobrushin and Major and yields a new example of a self-similar process with stationary increments.