Gaussian approximation for high dimensional time series
The Annals of Statistics · 2017 · Volume 45 · Issue 5
Abstract
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.
Keywords
62E17, 62M10, Gaussian approximation, high-dimensional time series, Kolmogorov–Smirnov test, long run covariance matrix, simultaneous inference
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