Wavelet Series Representation and Geometric Properties of Harmonizable Fractional Stable Sheets

Abstract: Let $Z^H= {Z^H(t), t \in \R<sup>N}$ be a real-valued $N$-parameter harmonizable fractional stable sheet with index $H = (H_1, \ldots, H_N) \in (0, 1)<sup>N$. We establish a random wavelet series expansion for $Z<sup>H$ which is almost surely convergent in all the H\"older spaces $C^\gamma ([-M,M]<sup>N)$, where $M&gt;0$ and $\gamma\in (0, \min{H_1,\ldots, H_N})$ are arbitrary. One of the main ingredients for proving the latter result is the LePage representation for a rotationally invariant stable random measure. Also, let $X={X(t), t \in \R<sup>N}$ be an $\R<sup>d$-valued harmonizable fractional stable sheet whose components are independent copies of $Z<sup>H$. By making essential use of the regularity of its local times, we prove that, on an event of positive probability, the formula for the Hausdorff dimension of the inverse image $X<sup>{-1}(F)$ holds for all Borel sets $F \subseteq \R<sup>d$. This is referred to as a uniform Hausdorff dimension result for the inverse images.