Sample path properties of generalized random sheets with operator scaling

TL;DR

This paper studies the regularity and fractal dimensions of operator scaling alpha-stable random sheets, extending previous models by combining operator scaling and fractional Brownian sheet properties.

Contribution

It establishes a uniform modulus of continuity and determines the box-counting and Hausdorff dimensions of the graph of these generalized random sheets.

Findings

Established a general uniform modulus of continuity.

Determined the box-counting dimension of the graph.

Calculated the Hausdorff dimension of the graph.

Abstract

We consider operator scaling $\alpha$-stable random sheets, which were introduced in [12]. The idea behind such fields is to combine the properties of operator scaling $\alpha$-stable random fields introduced in [6] and fractional Brownian sheets introduced in [14]. We establish a general uniform modulus of continuity of such fields in terms of the polar coordinates introduced [6]. Based on this, we determine the box-counting dimension and the Hausdorff dimension of the graph of a trajectory of such fields over a non-degenerate cube $I \subset Rd$.