  In this paper, we consider the estimation and inference of precision matrices
of a rich class of locally stationary and nonlinear time series assuming that
only one realization of the time series is observed. Using a Cholesky
decomposition technique, we show that the precision matrices can be directly
estimated via a series of least squares linear regressions with smoothly
time-varying coefficients. The method of sieves is utilized for the estimation
and is shown to be efficient and optimally adaptive in terms of estimation
accuracy and computational complexity. We establish an asymptotic theory for a
class of ${\cal L}^2$ tests based on the nonparametric sieve estimators. The
latter are used for testing whether the precision matrices are diagonal or
banded. A high dimensional Gaussian approximation result is established for a
wide class of quadratic form of non-stationary and nonlinear processes, which
is of interest by itself.
