Wavelet series representation and geometric properties of harmonizable fractional stable sheets

DOI10.1080/17442508.2019.1594811zbMATH Open1490.60126arXiv1903.04397OpenAlexW2950689656MaRDI QIDQ5086471

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Publication date: 5 July 2022

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Abstract: Let $Z^H=Z^H(t),tin{R}^N$ 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 $C^{g}amma([-M,M{]}^N)$, where $M>0$ and $gammain(0,min{H}_1,ldots,H_N)$ 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 $X=X(t),tin{R}^N$ 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 $Fsubseteq{R}^d$. This is referred to as a uniform Hausdorff dimension result for the inverse images.

Full work available at URL: https://arxiv.org/abs/1903.04397

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