Stationary increments harmonizable stable fields: upper estimates on path behaviour

TL;DR

This paper extends wavelet-based methods to analyze the sample path behavior of stationary increments harmonizable stable fields, overcoming challenges posed by heavy-tailed distributions and frequency domain complexities.

Contribution

It generalizes and improves wavelet strategies for studying sample paths from moving average stable processes to a broader harmonizable stable setting with stationary increments.

Findings

Developed new wavelet techniques for stable fields

Obtained upper estimates on path regularity

Extended analysis to general harmonizable stable processes

Abstract

Studying sample path behaviour of stochastic fields/processes is a classical research topic in probability theory and related areas such as fractal geometry. To this end, many methods have been developed since a long time in Gaussian frames. They often rely on some underlying "nice" Hilbertian structure, and can also require finiteness of moments of high order. Therefore, they can hardly be transposed to frames of heavy-tailed stable probability distributions. However, in the case of some linear non-anticipative moving average stable fields/processes, such as the linear fractional stable sheet and the linear multifractional stable motion, rather new wavelet strategies have already proved to be successful in order to obtain sharp moduli of continuity and other results on sample path behaviour. The main goal of our article is to show that, despite the difficulties inherent in the frequency domain, such kind of a wavelet methodology can be generalized and improved, so that it also becomes fruitful in a general harmonizable stable setting with stationary increments. Let us point out that there are large differences between this harmonizable setting and the moving average stable one.