  The existence of limiting spectral distribution (LSD) of
$\hat{\Gamma}_u+\hat{\Gamma}_u^*$, the symmetric sum of the sample
autocovariance matrix $\hat{\Gamma}_u$ of order $u$, is known when the
observations are from an infinite dimensional vector linear process with
appropriate (strong) assumptions on the coefficient matrices. Under
significantly weaker conditions, we prove, in a unified way, that the LSD of
any symmetric polynomial in these matrices such as
$\hat{\Gamma}_u+\hat{\Gamma}_u^*$, $\hat{\Gamma}_u\hat{\Gamma}_u^*$,
$\hat{\Gamma}_u\hat{\Gamma}_u^*+\hat{\Gamma}_k\hat{\Gamma}_k^*$ exist. Our
approach is through the more intuitive algebraic method of free probability in
conjunction with the method of moments. Thus, we are able to provide a general
description for the limits in terms of some freely independent variables. All
the previous results follow as special cases. We suggest statistical uses of
these LSD and related results in order determination and white noise testing.
