Wavelet series representation and geometric properties of harmonizable fractional stable sheets.

Let Z H = { Z H (t) , t ∈ R N } be a real-valued N-parameter harmonizable fractional stable sheet with index H = (H 1 , ... , H N) ∈ (0 , 1) N . We establish a random wavelet series expansion for Z H which is almost surely convergent in all the Hölder spaces C γ ([ − M , M ] N) , where M>0 and γ ∈ (0 , min { H 1 , ... , 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 ∈ 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 ⊆ R d . This is referred to as a uniform Hausdorff dimension result for the inverse images. [ABSTRACT FROM AUTHOR]