Noncentral limit theorem for the generalized Hermite process. (English) Zbl 1386.60190

Bell, Denis; Nualart, David

Electron. Commun. Probab. 22, Paper No. 66, 13 p. (2017).

Summary: 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 S. Bai and M. S. Taqqu [Ann. Probab. 45, No. 2, 1278–1324 (2017; Zbl 1392.60026)], who proved the result in the case \(q=2\) and without stability.

MSC:

60H05	Stochastic integrals
60H07	Stochastic calculus of variations and the Malliavin calculus
60F05	Central limit and other weak theorems

Keywords:

multiple stochastic integrals; Rosenblatt process; Hermite process; Skorokhod integral; central and noncentral limit theorems

Citations:

Zbl 1392.60026