ISSN:
1432-2064
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Summary We are given a random walk S 1, S 2, ... on ℤν, ν≧1, and a strongly correlated stationary random field ξ(x), xεℤν, 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-Itô 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00532965
