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October 2017 Gaussian approximation for high dimensional time series
Danna Zhang, Wei Biao Wu
Ann. Statist. 45(5): 1895-1919 (October 2017). DOI: 10.1214/16-AOS1512


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.


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Danna Zhang. Wei Biao Wu. "Gaussian approximation for high dimensional time series." Ann. Statist. 45 (5) 1895 - 1919, October 2017.


Received: 1 April 2016; Revised: 1 August 2016; Published: October 2017
First available in Project Euclid: 31 October 2017

zbMATH: 1381.62254
MathSciNet: MR3718156
Digital Object Identifier: 10.1214/16-AOS1512

Primary: 62M10
Secondary: 62E17

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

Rights: Copyright © 2017 Institute of Mathematical Statistics


Vol.45 • No. 5 • October 2017
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