Bernoulli

  • Bernoulli
  • Volume 24, Number 4A (2018), 2640-2675.

Gaussian approximation for high dimensional vector under physical dependence

Xianyang Zhang and Guang Cheng

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Abstract

We develop a Gaussian approximation result for the maximum of a sum of weakly dependent vectors, where the data dimension is allowed to be exponentially larger than sample size. Our result is established under the physical/functional dependence framework. This work can be viewed as a substantive extension of Chernozhukov et al. (Ann. Statist. 41 (2013) 2786–2819) to time series based on a variant of Stein’s method developed therein.

Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 2640-2675.

Dates
Received: January 2016
Revised: September 2016
First available in Project Euclid: 26 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1522051220

Digital Object Identifier
doi:10.3150/17-BEJ939

Mathematical Reviews number (MathSciNet)
MR3779697

Zentralblatt MATH identifier
06853260

Keywords
Gaussian approximation high dimensionality physical dependence measure Slepian interpolation Stein’s method time series

Citation

Zhang, Xianyang; Cheng, Guang. Gaussian approximation for high dimensional vector under physical dependence. Bernoulli 24 (2018), no. 4A, 2640--2675. doi:10.3150/17-BEJ939. https://projecteuclid.org/euclid.bj/1522051220


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References

  • [1] Andrews, D.W.K. (1984). Nonstrong mixing autoregressive processes. J. Appl. Probab. 21 930–934.
  • [2] Avram, F. and Bertsimas, D. (1993). On central limit theorems in geometrical probability. Ann. Appl. Probab. 3 1033–1046.
  • [3] Baldi, P. and Rinott, Y. (1989). Asymptotic normality of some graph-related statistics. J. Appl. Probab. 26 171–175.
  • [4] Baldi, P. and Rinott, Y. (1989). On normal approximations of distributions in terms of dependency graphs. Ann. Probab. 17 1646–1650.
  • [5] Bentkus, V. (2003). On the dependence of the Berry–Esseen bound on dimension. J. Statist. Plann. Inference 113 385–402.
  • [6] Cai, T.T. and Jiang, T. (2011). Limiting laws of coherence of random matrices with applications to testing covariance structure and construction of compressed sensing matrices. Ann. Statist. 39 1496–1525.
  • [7] Cai, T.T., Liu, W. and Xia, Y. (2014). Two-sample test of high dimensional means under dependence. J. R. Stat. Soc. Ser. B. Stat. Methodol. 76 349–372.
  • [8] Candes, E. and Tao, T. (2007). The Dantzig selector: Statistical estimation when $p$ is much larger than $n$. Ann. Statist. 35 2313–2351.
  • [9] Chatterjee, S. (2005). An error bound in the Sudakov–Fernique inequality. Preprint. Available at arXiv:math/0510424.
  • [10] Chen, X., Xu, M. and Wu, W.B. (2013). Covariance and precision matrix estimation for high-dimensional time series. Ann. Statist. 41 2994–3021.
  • [11] Chernozhukov, V., Chetverikov, D. and Kato, K. (2013). Testing many moment inequalities. Preprint. Available at arXiv:1312.7614.
  • [12] Chernozhukov, V., Chetverikov, D. and Kato, K. (2013). Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors. Ann. Statist. 41 2786–2819.
  • [13] Chernozhukov, V., Chetverikov, D. and Kato, K. (2015). Comparison and anti-concentration bounds for maxima of Gaussian random vectors. Probab. Theory Related Fields 162 47–70.
  • [14] Cho, H. and Fryzlewicz, P. (2014). Multiple change-point detection for high-dimensional time series via sparsified binary segmentation. J. R. Stat. Soc. Ser. B. Stat. Methodol. 77 475–507.
  • [15] de la Peña, V.H., Lai, T.L. and Shao, Q.-M. (2009). Self-Normalized Processes: Limit Theory and Statistical Applications. Probability and Its Applications (New York). Berlin: Springer.
  • [16] Götze, F. (1991). On the rate of convergence in the multivariate CLT. Ann. Probab. 19 724–739.
  • [17] Han, F. and Liu, H. (2014). A direct estimation of high dimensional stationary vector autoregressions. Preprint. Available at arXiv:1307.0293.
  • [18] Lam, C. and Yao, Q. (2012). Factor modeling for high-dimensional time series: Inference for the number of factors. Ann. Statist. 40 694–726.
  • [19] Liu, W. and Lin, Z. (2009). Strong approximation for a class of stationary processes. Stochastic Process. Appl. 119 249–280.
  • [20] Lütkepohl, H. (2005). New Introduction to Multiple Time Series Analysis. Berlin: Springer.
  • [21] Petrovskayam, B. and Leontovicha, M. (1982). The central limit theorem for a sequence of random variables with a slowly growing number of dependencies. Theory Probab. Appl. 27 815–825.
  • [22] Pinelis, I. (1994). Optimum bounds for the distributions of martingales in Banach spaces. Ann. Probab. 22 1679–1706.
  • [23] Portnoy, S. (1986). On the central limit theorem in $\mathbb{R} ^{p}$ when $p\rightarrow\infty$. Probab. Theory Related Fields 73 571–583.
  • [24] Röllin, A. (2013). Stein’s method in high dimensions with applications. Ann. Inst. Henri Poincaré Probab. Stat. 49 529–549.
  • [25] Romano, J. and Wolf, M. (2005). Exact and approximate stepdown methods for multiple hypothesis testing. J. Amer. Statist. Assoc. 100 94–108.
  • [26] Stein, C. (1986). Approximate Computation of Expectations. Institute of Mathematical Statistics Lecture Notes – Monograph Series 7. Hayward, CA: IMS.
  • [27] Wu, W.B. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA 102 14150–14154.
  • [28] Wu, W.B. (2011). Asymptotic theory for stationary processes. Stat. Interface 4 207–226.
  • [29] Wu, W.B. and Shao, X. (2004). Limit theorems for iterated random functions. J. Appl. Probab. 41 425–436.
  • [30] Wu, W.B. and Zhou, Z. (2011). Gaussian approximations for non-stationary multiple time series. Statist. Sinica 21 1397–1413.
  • [31] Zhang, D. and Wu, W.B. (2016). Gaussian approximation for high dimensional time series. Ann. Statist. To appear.
  • [32] Zhang, X. and Cheng, G. (2016). Simultaneous inference for high-dimensional linear models. J. Amer. Statist. Assoc. To appear.