• Bernoulli
  • Volume 21, Number 4 (2015), 2336-2350.

Density convergence in the Breuer–Major theorem for Gaussian stationary sequences

Yaozhong Hu, David Nualart, Samy Tindel, and Fangjun Xu

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Consider a Gaussian stationary sequence with unit variance $X=\{X_{k};k\in\mathbb{N}\cup\{0\}\}$. Assume that the central limit theorem holds for a weighted sum of the form $V_{n}=n^{-1/2}\sum^{n-1}_{k=0}f(X_{k})$, where $f$ designates a finite sum of Hermite polynomials. Then we prove that the uniform convergence of the density of $V_{n}$ towards the standard Gaussian density also holds true, under a mild additional assumption involving the causal representation of $X$.

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Bernoulli, Volume 21, Number 4 (2015), 2336-2350.

Received: March 2014
Revised: May 2014
First available in Project Euclid: 5 August 2015

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Breuer–Major theorem density convergence Gaussian stationary sequences Malliavin calculus moving average representation


Hu, Yaozhong; Nualart, David; Tindel, Samy; Xu, Fangjun. Density convergence in the Breuer–Major theorem for Gaussian stationary sequences. Bernoulli 21 (2015), no. 4, 2336--2350. doi:10.3150/14-BEJ646.

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  • [1] Arcones, M.A. (1994). Limit theorems for nonlinear functionals of a stationary Gaussian sequence of vectors. Ann. Probab. 22 2242–2274.
  • [2] Beran, J. (1994). Statistics for Long-Memory Processes. Monographs on Statistics and Applied Probability 61. New York: Chapman and Hall.
  • [3] Breuer, P. and Major, P. (1983). Central limit theorems for nonlinear functionals of Gaussian fields. J. Multivariate Anal. 13 425–441.
  • [4] Brockwell, P.J. and Davis, R.A. (1991). Time Series: Theory and Methods, 2nd ed. Springer Series in Statistics. New York: Springer.
  • [5] Carbery, A. and Wright, J. (2001). Distributional and $L^{q}$ norm inequalities for polynomials over convex bodies in $\mathbb{R}^{n}$. Math. Res. Lett. 8 233–248.
  • [6] Chambers, D. and Slud, E. (1989). Central limit theorems for nonlinear functionals of stationary Gaussian processes. Probab. Theory Related Fields 80 323–346.
  • [7] Giraitis, L. and Surgailis, D. (1985). CLT and other limit theorems for functionals of Gaussian processes. Z. Wahrsch. Verw. Gebiete 70 191–212.
  • [8] Granger, C.W.J. and Joyeux, R. (1980). An introduction to long-memory time series models and fractional differencing. J. Time Series Anal. 1 15–29.
  • [9] Gubner, J.A. (2005). Theorems and fallacies in the theory of long-range-dependent processes. IEEE Trans. Inform. Theory 51 1234–1239.
  • [10] Hosking, J.R.M. (1981). Fractional differencing. Biometrika 68 165–176.
  • [11] Hu, Y., Lu, F. and Nualart, D. (2014). Convergence of densities of some functionals of Gaussian processes. J. Funct. Anal. 266 814–875.
  • [12] Nourdin, I. and Nualart, D. (2013). Fisher information and the fourth moment theorem. Preprint. Available at arXiv:1312.5841.
  • [13] Nourdin, I. and Peccati, G. (2009). Stein’s method on Wiener chaos. Probab. Theory Related Fields 145 75–118.
  • [14] 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.
  • [15] Nourdin, I., Peccati, G. and Podolskij, M. (2011). Quantitative Breuer–Major theorems. Stochastic Process. Appl. 121 793–812.
  • [16] Nourdin, I. and Poly, G. (2013). Convergence in total variation on Wiener chaos. Stochastic Process. Appl. 123 651–674.
  • [17] Nualart, D. (2006). The Malliavin Calculus and Related Topics, 2nd ed. Probability and Its Applications (New York). Berlin: Springer.
  • [18] Nualart, D. and Peccati, G. (2005). Central limit theorems for sequences of multiple stochastic integrals. Ann. Probab. 33 177–193.
  • [19] Palma, W. (2007). Long-Memory Time Series: Theory and Methods. Wiley Series in Probability and Statistics. Hoboken, NJ: Wiley.