# Lower bound for local oscillations of Hermite processes

## TL;DR

This paper establishes a lower bound on the local oscillations of Hermite process paths, demonstrating that these non-Gaussian processes are almost surely nowhere differentiable, thus advancing understanding of their path regularity.

## Contribution

It provides the first quasi-optimal lower bound on local oscillations of Hermite processes of any rank, confirming their nowhere differentiability.

## Key findings

- Paths of Hermite processes are nowhere differentiable.
- The lower bound on local oscillations is quasi-optimal.
- Results apply to Hermite processes of any rank N.

## Abstract

The most known example of a class of non-Gaussian stochastic processes which belongs to the homogenous Wiener chaos of an arbitrary order N > 1 are probably Hermite processes of rank N. They generalize fractional Brownian motion (fBm) and Rosenblatt process in a natural way. They were introduced several decades ago. Yet, in contrast with fBm and many other Gaussian and stable stochastic processes and fields related to it, few results on path behavior of Hermite processes are available in the literature. For instance the natural issue of whether or not their paths are nowhere differentiable functions has not yet been solved even in the most simple case of the Rosenblatt process. The goal of our article is to derive a quasi-optimal lower bound of the asymptotic behavior of local oscillations of paths of Hermite processes of any rank N, which, among other things, shows that these paths are nowhere differentiable functions.