Wavelet Series Representation and Geometric Properties of Harmonizable Fractional Stable Sheets

TL;DR

This paper develops a wavelet series expansion for harmonizable fractional stable sheets, proving convergence in H"older spaces, and establishes a uniform Hausdorff dimension result for inverse images of these stochastic processes.

Contribution

It introduces a new wavelet series representation for harmonizable fractional stable sheets and proves a uniform Hausdorff dimension formula for their inverse images.

Findings

Wavelet series expansion converges in all H"older spaces.

Established Hausdorff dimension formula for inverse images.

Utilized LePage representation for stable random measures.

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.