High Dimensional Time Series and Free Probability 

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.