We present results about the limiting behavior of the empirical distribution of eigenvalues of a weighted integral of the sample periodogram for a class of high-dimensional linear processes. The processes under consideration are characterized by having simultaneously diagonalizable coefficient matrices. We make use of these asymptotic results, derived under the setting where the dimension and sample size are comparable, to formulate an estimation strategy for the distribution of eigenvalues of the coefficients of the linear process. This approach generalizes existing works on estimation of the spectrum of an unknown covariance matrix for high-dimensional i.i.d. observations. We also present an application of the proposed methodology to estimation of the mean variance frontier in the Markowitz portfolio optimization problem.