Wavelet Series Representation and Geometric Properties of Harmonizable Fractional Stable Sheets

Abstract

Let $Z^H= \{Z^H(t), t \in \R^N\}$ be a real-valued $N$-parameter harmonizable fractional stable sheet with index $H = (H_1, \ldots, H_N) \in (0, 1)^N$. We establish a random wavelet series expansion for $Z^H$ which is almost surely convergent in all the H\"older spaces $C^\gamma ([-M,M]^N)$, where $M>0$ 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 $X=\{X(t), t \in \R^N\}$ be an $\R^d$-valued harmonizable fractional stable sheet whose components are independent copies of $Z^H$. 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 $X^{-1}(F)$ holds for all Borel sets $F \subseteq \R^d$. This is referred to as a uniform Hausdorff dimension result for the inverse images.