Noncentral limit theorem for the generalized Rosenblatt process

TL;DR

This paper investigates the convergence behavior of generalized Rosenblatt processes using Malliavin calculus, revealing a stable convergence to a compound Gaussian distribution as parameters approach the boundary of their domain.

Contribution

It extends previous results by proving stable convergence of generalized Rosenblatt processes to a compound Gaussian distribution in a multi-parameter setting.

Findings

Convergence to a compound Gaussian distribution as parameters approach the boundary of the domain.

Stable convergence in law of the generalized Rosenblatt processes.

Generalization of previous $q=2$ case to higher dimensions.

Abstract

We use techniques of Malliavin calculus to study the convergence in law of a family of generalized Rosenblatt processes $Z_\gamma$ with kernels defined by parameters $\gamma$ taking values in a tetrahedral region $\Delta$ of $\RR^q$. We prove that, as $\gamma$ converges to a face of $\Delta$, the process $Z_\gamma$ converges to a compound Gaussian distribution with random variance given by the square of a Rosenblatt process of one lower rank. The convergence in law is shown to be stable. This work generalizes a previous result of Bai and Taqqu, who proved the result in the case $q=2$ and without stability.