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.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ244 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Generalized dimensions of images of measures under Gaussian processes

“…For more general Borel measures µ, one can look at the almost sure Hausdorff and packing dimensions of the measures (given by the minimal dimension of any set with complement of zero measure); indeed, by supporting suitable measures on sets, this approach is often implicit in the set dimension results mentioned above. Explicit results for Hausdorff and packing dimensions of image measures under a wide range of processes are given in [29], with dimension profiles again key in the packing dimension cases.…”

On the carrying dimension of occupation measures for self-affine random fields

“…Hence an approach analogous to Theorem 3.1 for the graph, combining the Hausdorff dimension of the range with the carrying dimension of σ X , fails; cf. also [44]. Nevertheless, if the carrying dimension of σ X exists, then obviously cardim σ X ≤ cardim τ X .…”

Fractal dimension results for continuous time random walks

“…Proposition 2.1 and Theorem 2.2 are applicable here, but the Hausdorff dimension of the range and graph sets of the processes Y and Z ( u ) = ( D ( u ), Y ( u )) are unknown in general. For some partial results in this direction, see Shieh and Xiao (2010) and Xiao and Lin (1994).…”