The Annals of Statistics

Gaussian approximation for high dimensional time series

Danna Zhang and Wei Biao Wu

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We consider the problem of approximating sums of high dimensional stationary time series by Gaussian vectors, using the framework of functional dependence measure. The validity of the Gaussian approximation depends on the sample size $n$, the dimension $p$, the moment condition and the dependence of the underlying processes. We also consider an estimator for long-run covariance matrices and study its convergence properties. Our results allow constructing simultaneous confidence intervals for mean vectors of high-dimensional time series with asymptotically correct coverage probabilities. As an application, we propose a Kolmogorov–Smirnov-type statistic for testing distributions of high-dimensional time series.

Article information

Ann. Statist., Volume 45, Number 5 (2017), 1895-1919.

Received: April 2016
Revised: August 2016
First available in Project Euclid: 31 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62E17: Approximations to distributions (nonasymptotic)

Gaussian approximation high-dimensional time series Kolmogorov–Smirnov test long run covariance matrix simultaneous inference


Zhang, Danna; Wu, Wei Biao. Gaussian approximation for high dimensional time series. Ann. Statist. 45 (2017), no. 5, 1895--1919. doi:10.1214/16-AOS1512.

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Supplemental materials

  • Supplement to “Gaussian approximation for high dimensional time series”. This supplemental file contains the additional technical proofs and a simulation study.