Joint convergence of sample autocovariance matrices when $p/n\to 0$ with application

Consider a high dimensional linear time series model where the dimension p and the sample size n grow in such a way that p/n → 0. Let Γ̂u be the uth order sample autocovariance matrix. We first show that the LSD of any symmetric polynomial in {Γ̂u, Γ̂u, u ≥ 0} exists under independence and moment assumptions on the driving sequence together with weak assumptions on the coefficient matrices. This LSD result, with some additional effort, implies the asymptotic normality of the trace of any polynomial in {Γ̂u, Γ̂u, u ≥ 0}. We also study similar results for several independent MA processes. We show applications of the above results to statistical inference problems such as in estimation of the unknown order of a highdimensional MA process and in graphical and significance tests for hypotheses on coefficient matrices of one or several such independent processes.