Wavelet series representation and geometric properties of harmonizable fractional stable sheets

ABSTRACT

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

KEYWORDS:

• Harmonizable fractional stable sheets
• wavelet series representation
• LePage representation
• strong local nondeterminism
• local times
• Hausdorff dimension
• inverse image

2000 AMS CLASSIFICATION NUMBERS:

• 60G52
• 60G17
• 60G18
• 60G60

Acknowledgments

This work was finished during Yimin Xiao's visit to Université de Lille sponsored by CEMPI (ANR-11-LABX-0007-01). The hospitality from this university and the financial support from CEMPI are appreciated.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of Yimin Xiao is partially supported by NSF of USA grants DMS-1612885 and DMS-1607089.