  We show that the empirical spectral distribution (ESD) of the sample
autocovariance matrix (ACVM) converges as the dimension increases, when the
time series is a linear process with reasonable restriction on the
coefficients. The limit does not depend on the distribution of the underlying
driving i.i.d. sequence and its support is unbounded. This limit does not
coincide with the spectral distribution of the theoretical ACVM. However, it
does so if we consider a suitably tapered version of the sample ACVM. For
banded sample ACVM the limit has unbounded support as long as the number of
non-zero diagonals in proportion to the dimension of the matrix is bounded away
from zero. If this ratio tends to zero, then the limit exists and again
coincides with the spectral distribution of the theoretical ACVM. Finally, we
also study the LSD of a naturally modified version of the ACVM which is not
non-negative definite.
