Linear Fractional Stable Sheets: wavelet expansion and sample path properties

TL;DR

This paper provides a detailed wavelet series representation of linear fractional stable sheets, analyzing their sample path properties, continuity, and fractal dimensions, thereby advancing understanding of their geometric and probabilistic behavior.

Contribution

It introduces a wavelet expansion for linear fractional stable sheets and derives their path regularity and fractal dimensions, extending prior theoretical results.

Findings

Wavelet series representation of stable sheets

Established modulus of continuity for sample paths

Calculated Hausdorff dimensions of range and graph

Abstract

In this paper we give a detailed description of the random wavelet series representation of real-valued linear fractional stable sheet introduced in Ayache, Roueff and Xiao (2007). By using this representation, in the case where the sample paths are continuous, an anisotropic uniform and quasi-optimal modulus of continuity of these paths is obtained as well as an upper bound for their behavior at infinity and around the coordinate axes. The Hausdorff dimensions of the range and graph of these stable random fields are then derived.