## Electronic Communications in Probability

### 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.

#### Article information

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

Dates
Accepted: 13 November 2017
First available in Project Euclid: 23 November 2017

https://projecteuclid.org/euclid.ecp/1511427621

Digital Object Identifier
doi:10.1214/17-ECP99

Mathematical Reviews number (MathSciNet)
MR3734105

Zentralblatt MATH identifier
06827048

#### Citation

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

#### References

• [1] Bai, S. and Taqqu. M.: Structure of the third moment of the generalized Rosenblatt distribution. Stoch. Proc. Appl. 124, (2014), 1710–1739.
• [2] Bai, S. and Taqqu, M.: Behavior of the generalized Rosenblatt process at extreme critical exponent values. Ann. Probab. 45, (2017), 1278–1324.
• [3] Bell, D.: The Malliavin calculus. Dover Publications, Inc., Mineola, NY, 2006. x+135 pp.
• [4] Dobrushin, R. L. and Major, P.: Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50, (1979), 27–52.
• [5] Maejima, M. and Tudor, C. A. : Wiener integrals with respect to the Hermite process and a non-central limit theorem. Stoch. Anal. Appl. 25, (2007), 1043–1056.
• [6] Maejima, M. and Tudor, C. A.: Selfsimilar processes with stationary increments in the second Wiener chaos. Probability and Mathematical Statistics 32, (2012), 167–186.
• [7] Nourdin, I. and Nualart, D.: Central limit theorems for multiple Skorohod integrals. J. Theoret. Probab. 23, (2010) 39–64.
• [8] Nourdin, I., Nualart, D. and Peccati, G.: Quantitative stable limit theorems on the Wiener space. Ann. Probab. 44, (2016), 1–41.
• [9] Nourdin, I. and Peccati, G.: Normal Approximations with Malliavin Calculus. From Stein’s Method to Universality. Cambridge University Press, 2012. xiv+239 pp.
• [10] Nourdin, I. and Peccati, G.: The optimal fourth moment theorem. Proc. Amer. Math. Soc. 143, (2015), 3123–3133.
• [11] Nualart, D.: The Malliavin calculus and related topics. Second edition. Springer-Verlag, Berlin, 2006. xiv+382 pp.
• [12] Nualart, D. and Peccati, D.: Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33, (2005), 177–193.
• [13] Skorohod, A. V.: On a generalization of a stochastic integral. Theory Probab. Appl. 20, (1975), 223–238.
• [14] Taqqu, M.: Weak convergence to fractional Brownian motion and to the Rosenblatt process. Probab. Theory Rel. Fields 31, (1974/75), 287–302.
• [15] Taqqu, M.: Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 50, (1979), 53–83.