On a class of self-similar processes with stationary increments in higher order Wiener chaoses

TL;DR

This paper investigates a class of self-similar processes with stationary increments in higher order Wiener chaoses, providing wavelet-like expansions, regularity analysis, and Hausdorff dimension results for their sample paths.

Contribution

It introduces a wavelet-like expansion for these processes and analyzes their regularity and fractal properties, extending understanding of higher order Wiener chaos processes.

Findings

Almost sure wavelet-like expansion derived

Pointwise and local Hölder regularity characterized

Hausdorff dimension of range and graphs analyzed

Abstract

We study a class of self-similar processes with stationary increments belonging to higher order Wiener chaoses which are similar to Hermite processes. We obtain an almost sure wavelet-like expansion of these processes. This allows us to compute the pointwise and local H\"older regularity of sample paths and to analyse their behaviour at infinity. We also provide some results on the Hausdorff dimension of the range and graphs of multidimensional anisotropic self-similar processes with stationary increments defined by multiple Wiener integrals.