Hausdorff and packing dimensions of the images of random fields

TL;DR

This paper investigates the Hausdorff and packing dimensions of images of various random fields, providing formulas and conditions applicable to Gaussian, stable, and other complex stochastic processes.

Contribution

It derives general results for the dimensions of image measures and sets of diverse random fields under mild conditions, extending previous work to new classes of processes.

Findings

Formulas for Hausdorff and packing dimensions of image measures and sets.

Applicable to Gaussian, stable, and fractional Lévy fields.

Provides conditions under which these dimensions are determined.

Abstract

Let $X=\{X(t),t\in\mathbb{R}^N\}$ be a random field with values in $\mathbb{R}^d$. For any finite Borel measure $\mu$ and analytic set $E\subset\mathbb{R}^N$, the Hausdorff and packing dimensions of the image measure $\mu_X$ and image set $X(E)$ are determined under certain mild conditions. These results are applicable to Gaussian random fields, self-similar stable random fields with stationary increments, real harmonizable fractional L\'{e}vy fields and the Rosenblatt process.