  This article is concerned with the spectral behavior of $p$-dimensional
linear processes in the moderately high-dimensional case when both
dimensionality $p$ and sample size $n$ tend to infinity so that $p/n\to0$. It
is shown that, under an appropriate set of assumptions, the empirical spectral
distributions of the renormalized and symmetrized sample autocovariance
matrices converge almost surely to a nonrandom limit distribution supported on
the real line. The key assumption is that the linear process is driven by a
sequence of $p$-dimensional real or complex random vectors with i.i.d. entries
possessing zero mean, unit variance and finite fourth moments, and that the
$p\times p$ linear process coefficient matrices are Hermitian and
simultaneously diagonalizable. Several relaxations of these assumptions are
discussed. The results put forth in this paper can help facilitate inference on
model parameters, model diagnostics and prediction of future values of the
linear process.
