[Paper Review] Sample path properties of multivariate operator-self-similar stable random fields

Ercan Sönmez|arXiv (Cornell University)|Feb 3, 2016

Mathematical Dynamics and Fractals22 references3 citations

TL;DR

This paper establishes sample path regularity and computes the Hausdorff dimension of multivariate operator-self-similar $α$-stable random fields via a harmonizable representation. By leveraging results from [8], it derives an upper bound on the modulus of continuity, resolving an open problem in [25] regarding the dimension of sample paths.

ABSTRACT

We investigate the sample path regularity of multivariate operator-self-similar {\alpha}-stable random fields with values in R m given by a harmonizable representation. Such fields were introduced in [25] as a generalization of both operator-self-similar stochastic processes and operator scaling random fields and satisfy the scaling property X(c E t) =c D X(t), where E is a real d times d matrix and D is a real m times m matrix. By using results in [8] we give an upper bound on the modulus of continuity. Based on this we determine the Hausdorff dimension of the sample paths. In particular, this solves an open problem in [25].

Motivation & Objective

• To analyze the sample path regularity of multivariate operator-self-similar $α$-stable random fields with values in $\mathbb{R}^m$.
• To resolve an open problem concerning the Hausdorff dimension of sample paths in such fields, as posed in [25].
• To establish an upper bound on the modulus of continuity for these random fields using existing theoretical results.

Proposed method

• Utilizing a harmonizable representation to define the multivariate operator-self-similar $\alpha$-stable random fields.
• Applying results from [8] to derive an upper bound on the modulus of continuity of the sample paths.
• Employing the derived continuity bound to analyze the fractal geometry of the sample paths.
• Using the modulus of continuity estimate to compute the Hausdorff dimension of the sample paths.

Experimental results

Research questions

• RQ1 What is the sample path regularity of multivariate operator-self-similar $\alpha$-stable random fields with values in $\mathbb{R}^m$?
• RQ2 How does the scaling property $X(c^E t) = c^D X(t)$ influence the path behavior of these fields?
• RQ3 What is the Hausdorff dimension of the sample paths of such fields, and how can it be computed?
• RQ4 Can the open problem on path dimension in [25] be resolved using continuity bounds?

Key findings

• An upper bound on the modulus of continuity of the sample paths is established using theoretical results from [8].
• The Hausdorff dimension of the sample paths is computed based on the derived continuity bound.
• The study resolves an open problem in [25] concerning the dimension of sample paths in multivariate operator-self-similar $\alpha$-stable fields.
• The results confirm that the path regularity and geometric dimension are governed by the scaling matrices $E$ and $D$.