Consider a linear time series model whose dimension \$p\$

grows with the sample size \$n\$. We assume that \$p/n\$ goes to a finite

\$y\$. We show the large sample bulk eigenvalue properties of the sample

autocovariance matrices, including those of their polynomial

functions, and also show the joint convergence of these matrices in an

algebraic sense. Indeed we can put these ideas in a broadened

framework and work out the bulk behavior of matrix polynomials of

several sample variance-covariance matrices and deterministic

matrices. The proofs use ideas from Random Matrix Theory and Free Probability, including

properties of large dimensional IID and Wigner matrices.

The limits are identified as certain functions of free variables.

These results can be applied to statistical inference problems in high

dimensional linear time series models.

For various hypotheses, graphical tests based on the nature of the

limits, as well as significance tests based on the asymptotic

normality of the traces can be developed.