Consider a high dimensional linear time series model where both the dimension p and the sample size n grow indefinitely. We show that any finite degree matrix polynomial of a fixed number of autocovariance matrices converges tracially under independence and moment assumptions on the driving sequence together with weak assumptions on the coefficient matrices. In addition, if the polynomial is a symmetric matrix, then its empirical eigenvalue distribution converges weakly This LSD result, with some additional effort, implies the asymptotic normality of the trace of any polynomial. We also study similar results for several independent liner processes. We then discuss some of the difficulties involved in establishing the latter limits for non-symmetric polynomials.
We show applications of the above results to statistical inference problems such as in estimation of the unknown order of a high-dimensional MA process and in graphical and significance tests for hypotheses on coefficient matrices of one or several such independent processes.
This is joint work with Monika Bhattacharjee and Walid Hachem.