An Optimal Uniform Modulus of Continuity for Harmonizable Fractional Stable Motion

TL;DR

This paper determines the optimal logarithmic factor in the uniform modulus of continuity for Harmonizable Fractional Stable Motion, advancing understanding of its regularity properties and solving a long-standing open problem.

Contribution

It identifies the precise power of the logarithmic factor in the modulus of continuity for HFSM, using Abel transforms of LePage series expansions.

Findings

Established the optimal logarithmic power in the modulus of continuity.

Extended methodology to more general harmonizable stable processes.

Solved a long-standing open problem in the regularity of HFSM.

Abstract

Non-Gaussian Harmonizable Fractional Stable Motion (HFSM) is a natural and important extension of the well-known Fractional Brownian Motion to the framework of heavy-tailed stable distributions. It was introduced several decades ago; however its properties are far from being completely understood. In our present paper we determine the optimal power of the logarithmic factor in a uniform modulus of continuity for HFSM, which solves an open old problem. The keystone of our strategy consists in Abel transforms of the LePage series expansions of the random coefficients of the wavelets series representation of HFSM. Our methodology can be extended to more general harmonizable stable processes and fields.