Article
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Fractional Operators and Fractionally Integrated Random Fields on Z
Version 1
: Received: 17 May 2024 / Approved: 17 May 2024 / Online: 21 May 2024 (10:34:48 CEST)
How to cite: Surgailis, D.; Pilipauskaitė, V. Fractional Operators and Fractionally Integrated Random Fields on Z. Preprints 2024, 2024051180. https://doi.org/10.20944/preprints202405.1180.v1 Surgailis, D.; Pilipauskaitė, V. Fractional Operators and Fractionally Integrated Random Fields on Z. Preprints 2024, 2024051180. https://doi.org/10.20944/preprints202405.1180.v1
Abstract
We consider fractional integral operators $(I-T)^d, d \in (-1,1)$ acting on functions $g: \mathbb{Z}^{\nu} \to \mathbb{R}, \nu \ge 1 $, where $T $ is the transition operator of a random walk on
$\mathbb{Z}^{\nu}$. We obtain sufficient and necessary conditions for the existence, invertibility and square summability of kernels $\tau (\mathbf{s}; d), \mathmb{s} \in \mathbb{Z}^{\nu}$ of $(I-T)^d $. Asymptotic behavior of $\tau (\mathbf{s}; d)$ as $|\mathbf{s}| \to \infty$ is identified following local limit theorem for random walk.
A class of fractionally integrated random fields $X$ on $\mathbb{Z}^{\nu}$ solving the difference equation $(I-T)^d X = \varepsilon$ 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.
Keywords
fractional differentiation/integration operators; tempered fractional operators; fractional random field; random walk; limit theorems; long-range dependence; negative dependence; conditional autoregression
Subject
Computer Science and Mathematics, Probability and Statistics
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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