# Limit behavior of the Rosenblatt Ornstein-Uhlenbeck process with respect to the Hurst index

## TL;DR

This paper investigates the limiting behavior of the Rosenblatt Ornstein-Uhlenbeck process as the Hurst index approaches 1/2 and 1, revealing how the process converges in distribution in these regimes.

## Contribution

It provides a detailed analysis of the distributional limits of integrals involving the Rosenblatt process, specifically for the Ornstein-Uhlenbeck case, as the Hurst parameter varies.

## Key findings

- Convergence in distribution as H approaches 1/2 and 1.
- Characterization of the limit processes for different H values.
- Insights into the behavior of the Rosenblatt Ornstein-Uhlenbeck process.

## Abstract

We study the convergence in distribution, as $H\to \frac{1}{2}$ and as $H\to 1$, of the integral $\int_{\mathbb{R}} f(u) dZ^{H}(u) $, where $Z ^{H}$ is a Rosenblatt process with self-similarity index $H\in \left( \frac{1}{2}, 1\right) $ and $f$ is a suitable deterministic function. We focus our analysis on the case of the Rosenblatt Ornstein-Uhlenbeck process, which is the solution of the Langevin equation driven by the Rosenblatt process.