Electronic Communications in Probability

Noncentral limit theorem for the generalized Hermite process

Denis Bell and David Nualart

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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.

Article information

Electron. Commun. Probab., Volume 22 (2017), paper no. 66, 13 pp.

Received: 5 May 2017
Accepted: 13 November 2017
First available in Project Euclid: 23 November 2017

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Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals 60H07: Stochastic calculus of variations and the Malliavin calculus 60F05: Central limit and other weak theorems 65G18

multiple stochastic integrals Rosenblatt process Hermite process Skorohod integral central and noncentral limit theorems.

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Bell, Denis; Nualart, David. Noncentral limit theorem for the generalized Hermite process. Electron. Commun. Probab. 22 (2017), paper no. 66, 13 pp. doi:10.1214/17-ECP99. https://projecteuclid.org/euclid.ecp/1511427621

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