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

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.