Noncentral limit theorem for the generalized Hermite process

Abstract

We use techniques of Malliavin calculus to study the convergence in law of a family of generalized Hermite processes $Z_\gamma $ with kernels defined by parameters $\gamma $ taking values in a tetrahedral region $\Delta $ of $\mathbb{R} ^q$. 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 Hermite 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.