Noncentral limit theorem for the generalized Hermite process

Denis Bell,David Nualart

doi:10.1214/17-ecp99

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.

Highlights

Let W = {Wx, x ∈ R} be a two-sided Brownian motion on the real line
We use techniques of Malliavin calculus to study the convergence in law of a family of generalized Hermite processes Zγ with kernels defined by parameters γ taking values in a tetrahedral region ∆ of Rq
As γ converges to a face of ∆, the process Zγ converges to a compound Gaussian distribution with random variance given by the square of a Hermite process of one lower rank

Summary

Introduction

The goal of this paper is to derive this result as an application of a general theorem of convergence in law of multiple stochastic integrals to a mixture of Gaussian distributions (see Theorem 3.2), which is of independent interest This theorem is proved using a noncentral limit theorem for Skorohod integrals derived by Nourdin and Nualart in [7]. This allows us to extend Bai and Taqqu’s result in two directions: We can deal with a general Hermite in the qth Wiener chaos, and we can show that the convergence is stable.

Multiple stochastic integrals

Elements of Malliavin calculus

Noncentral limit theorems for multiple stochastic integrals

Generalized Hermite process