Hausdorff and packing dimensions of the images of random fields

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.