GAUSSIAN APPROXIMATION FOR HIGH DIMENSIONAL TIME SERIES

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...

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Bibliographic Details
Published in:The Annals of statistics 2017-10, Vol.45 (5), p.1895-1919
Main Authors: Zhang, Danna, Wu, Wei Biao
Format: Article
Language:English
Subjects:
Approximation
Confidence intervals
Covariance matrix
Mathematical analysis
Normal distribution
Statistical analysis
Studies
Time series
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Summary: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.
ISSN:0090-5364
2168-8966
DOI:10.1214/16-aos1512