• Bernoulli
  • Volume 23, Number 1 (2017), 710-742.

The impact of the diagonals of polynomial forms on limit theorems with long memory

Shuyang Bai and Murad S. Taqqu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We start with an i.i.d. sequence and consider the product of two polynomial-forms moving averages based on that sequence. The coefficients of the polynomial forms are asymptotically slowly decaying homogeneous functions so that these processes have long memory. The product of these two polynomial forms is a stationary nonlinear process. Our goal is to obtain limit theorems for the normalized sums of this product process in three cases: exclusion of the diagonal terms of the polynomial form, inclusion, or the mixed case (one polynomial form excludes the diagonals while the other one includes them). In any one of these cases, if the product has long memory, then the limits are given by Wiener chaos. But the limits in each of the cases are quite different. If the diagonals are excluded, then the limit is expressed as in the product formula of two Wiener–Itô integrals. When the diagonals are included, the limit stochastic integrals are typically due to a single factor of the product, namely the one with the strongest memory. In the mixed case, the limit stochastic integral is due to the polynomial form without the diagonals irrespective of the strength of the memory.

Article information

Bernoulli, Volume 23, Number 1 (2017), 710-742.

Received: March 2014
Revised: November 2014
First available in Project Euclid: 27 September 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

diagonals long memory noncentral limit theorem self-similar processes Volterra Wiener


Bai, Shuyang; Taqqu, Murad S. The impact of the diagonals of polynomial forms on limit theorems with long memory. Bernoulli 23 (2017), no. 1, 710--742. doi:10.3150/15-BEJ697.

Export citation


  • [1] Bai, S. and Taqqu, M.S. (2014). Generalized Hermite processes, discrete chaos and limit theorems. Stochastic Process. Appl. 124 1710–1739.
  • [2] Bai, S. and Taqqu, M.S. (2015). Convergence of long-memory discrete $k$th order Volterra processes. Stochastic Process. Appl. 125 2026–2053.
  • [3] Beran, J., Feng, Y., Ghosh, S. and Kulik, R. (2013). Long-Memory Processes: Probabilistic Properties and Statistical Methods. Heidelberg: Springer.
  • [4] Breton, J.-C. (2006). Convergence in variation of the joint laws of multiple Wiener–Itô integrals. Statist. Probab. Lett. 76 1904–1913.
  • [5] Davydov, Yu.A. and Martynova, G.V. (1989). Limit behavior of distributions of multiple stochastic integrals. In Statistics and Control of Random Processes (Russian) (Preila, 1987) 55–57. Moscow: “Nauka”.
  • [6] Dobrushin, R.L. and Major, P. (1979). Non-central limit theorems for nonlinear functionals of Gaussian fields. Z. Wahrsch. Verw. Gebiete 50 27–52.
  • [7] Embrechts, P. and Maejima, M. (2002). Selfsimilar Processes. Princeton Series in Applied Mathematics. Princeton, NJ: Princeton Univ. Press.
  • [8] Giraitis, L., Koul, H.L. and Surgailis, D. (2012). Large Sample Inference for Long Memory Processes. London: Imperial College Press.
  • [9] Hu, Y.Z. and Meyer, P.-A. (1988). Sur les intégrales multiples de Stratonovitch. In Séminaire de Probabilités, XXII. Lecture Notes in Math. 1321 72–81. Berlin: Springer.
  • [10] Koul, H.L., Baillie, R.T. and Surgailis, D. (2004). Regression model fitting with a long memory covariate process. Econometric Theory 20 485–512.
  • [11] Maejima, M. and Tudor, C.A. (2012). Selfsimilar processes with stationary increments in the second Wiener chaos. Probab. Math. Statist. 32 167–186.
  • [12] Nourdin, I. and Peccati, G. (2012). Normal Approximations with Malliavin Calculus: From Stein’s Method to Universality. Cambridge Tracts in Mathematics 192. Cambridge: Cambridge Univ. Press.
  • [13] Nourdin, I., Peccati, G. and Reinert, G. (2010). Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos. Ann. Probab. 38 1947–1985.
  • [14] Nourdin, I. and Poly, G. (2013). Convergence in total variation on Wiener chaos. Stochastic Process. Appl. 123 651–674.
  • [15] Surgailis, D. (1982). Domains of attraction of self-similar multiple integrals. Litovsk. Mat. Sb. 22 185–201.
  • [16] Taqqu, M.S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrsch. Verw. Gebiete 50 53–83.