Wavelet series representation and geometric properties of harmonizable fractional stable sheets

Abstract

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.

Hölder regularity of a Wiener integral in abstract space

“…]), wavelet theory (e.g. Ayache et al [4]), regular parametrization of random curves (e.g. Lawler and Zhou [28]) and many more.…”