We obtain a sharp convergence rate for banded covariance matrix estimates of
stationary processes. A precise order of magnitude is derived for spectral
radius of sample covariance matrices. We also consider thresholded covariance
matrix estimator that can better characterize sparsity if the true covariance
matrix is sparse. As our main tool, we implement Toeplitz (1911)'s idea and
relate eigenvalues of covariance matrices to the spectral densities or Fourier
transforms of the covariances.

We consider kernel estimation of marginal densities and regression functions
of stationary processes. It is shown that for a wide class of time series, with
proper centering and scaling, the maximum deviations of kernel density and
regression estimates are asymptotically Gumbel. Our results substantially
generalize earlier ones which were obtained under independence or beta mixing
assumptions.

This paper considers the efficient estimation of copula-based semiparametric
strictly stationary Markov models. These models are characterized by
nonparametric invariant (one-dimensional marginal) distributions and parametric
bivariate copula functions where the copulas capture temporal dependence and
tail dependence of the processes. The Markov processes generated via tail
dependent copulas may look highly persistent and are useful for financial and
economic applications.

For statistical inference of means of stationary processes, one needs to
estimate their time-average variance constants (TAVC) or long-run variances.
For a stationary process, its TAVC is the sum of all its covariances and it is
a multiple of the spectral density at zero. The classical TAVC estimate which
is based on batched means does not allow recursive updates and the required
memory complexity is O(n). We propose a faster algorithm which recursively
computes the TAVC, thus having memory complexity of order O(1) and the
computational complexity scales linearly in $n$.