Malliavin smoothness of the Rosenblatt process

TL;DR

This work addresses the smoothness of densities for finite-dimensional distributions of the Rosenblatt process, a non-Gaussian Hermite process in the second Wiener chaos. By leveraging Malliavin calculus and the Bouleau–Hirsch criterion, the authors establish nondegeneracy for Rosenblatt vectors through explicit control of negative moments of the Malliavin determinant, exploiting the second-chaos structure as a weighted sum of independent centered chi-square variables. They prove that increments and, consequently, Rosenblatt vectors admit densities in the Schwartz space $ ablaig( ablaig)$, and they derive exponential-type bounds for all partial derivatives of these densities. The results provide sharp, increment-dependent tail estimates and regularity information with potential applications to statistical inference for long-range dependent non-Gaussian processes.

Abstract

We investigate the smoothness of the densities of the finite-dimensional distributions of the Rosenblatt process. Within the Malliavin calculus framework, we prove that Rosenblatt random vectors are nondegenerate in the Malliavin sense. As a consequence, their densities belong to the Schwartz space of rapidly decreasing smooth functions. The proof relies on establishing the existence of all negative moments of the determinant of the Malliavin matrix, exploiting the specific structure of random variables in the second Wiener chaos. In addition, we derive exponential-type upper bounds for the partial derivatives of the densities of the finite-dimensional distributions of the Rosenblatt process.

Theorems & Definitions (15)

• Theorem 1
• Theorem 2
• Definition 1
• Lemma 1
• Theorem 3
• Lemma 2
• Theorem 4
• Lemma 3
• Proposition 1
• proof