Affiliation:
1. Department of Mathematical Sciences, Aalborg University, Skjernvej 4A, 9220 Aalborg, Denmark
2. Faculty of Mathematics and Informatics, Vilnius University, Naugarduko 24, 03225 Vilnius, Lithuania
Abstract
We consider fractional integral operators (I−T)d,d∈(−1,1) acting on functions g:Zν→R,ν≥1, where T is the transition operator of a random walk on Zν. We obtain the sufficient and necessary conditions for the existence, invertibility, and square summability of kernels τ(s;d),s∈Zν of (I−T)d. The asymptotic behavior of τ(s;d) as |s|→∞ is identified following the local limit theorem for random walks. A class of fractionally integrated random fields X on Zν solving the difference equation (I−T)dX=ε with white noise on the right-hand side is discussed and their scaling limits. Several examples, including fractional lattice Laplace and heat operators, are studied in detail.
Reference35 articles.
1. Brockwell, P.J., and Davis, R.A. (1991). Time Series: Theory and Methods, Springer.
2. Doukhan, P., Oppenheim, G., and Taqqu, M.S. (2003). Theory and Applications of Long-Range Dependence, Birkhäuser.
3. Giraitis, L., Koul, H.L., and Surgailis, D. (2012). Large Sample Inference for Long Memory Processes, Imperial College Press.
4. Pipiras, V., and Taqqu, M.S. (2017). Long-Range Dependence and Self-Similarity, Cambridge University Press.
5. Samorodnitsky, G., and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes, Chapman and Hall.
Cited by
1 articles.
订阅此论文施引文献
订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献