Statistics and Its Interface

Volume 4 (2011)

Number 2

Asymptotic theory for stationary processes

Pages: 207 – 226



Wei Biao Wu (Department of Statistics, The University of Chicago, Chicago, Il., U.S.A.)


We present a systematic asymptotic theory for statistics of stationary time series. In particular, we consider properties of sample means, sample covariance functions, covariance matrix estimates, periodograms, spectral density estimates, $U$-statistics, kernel density and regression estimates of linear and nonlinear processes. The asymptotic theory is built upon physical and predictive dependence measures, a new measure of dependence which is based on nonlinear system theory. Our dependence measures are particularly useful for dealing with complicated statistics of time series such as eigenvalues of sample covariance matrices and maximum deviations of nonparametric curve estimates.


dependence, covariance function, covariance matrix estimation, periodogram, spectral density estimation, U-statistics, kernel estimation, invariance principle, nonlinear time series

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