Electronic Communications in Probability

Noncentral limit theorem for the generalized Hermite process

Denis Bell and David Nualart

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1511427621

Digital Object Identifier
doi:10.1214/17-ECP99

Mathematical Reviews number (MathSciNet)
MR3734105

Zentralblatt MATH identifier
06827048

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

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

Rights
Creative Commons Attribution 4.0 International License.

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


Export citation

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.