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Large sample behaviour of high dimensional autocovariance matrices

The existence of limiting spectral distribution (LSD) of $\hatΓ_u+\hatΓ_u^*$, the symmetric sum of the sample autocovariance matrix $\hatΓ_u$ of order $u$, is known when the observations are from an infinite dimensional vector linear process with appropriate (strong) assumptions on the coefficient matrices. Under significantly weaker conditions, we prove, in a unified way, that the LSD of any symmetric polynomial in these matrices such as $\hatΓ_u+\hatΓ_u^*$, $\hatΓ_u\hatΓ_u^*$, $\hatΓ_u\hatΓ_u^*+\hatΓ_k\hatΓ_k^*$ exist. Our approach is through the more intuitive algebraic method of free probability in conjunction with the method of moments. Thus, we are able to provide a general description for the limits in terms of some freely independent variables. All the previous results follow as special cases. We suggest statistical uses of these LSD and related results in order determination and white noise testing.