Large sample behaviour of high dimensional autocovariance matrices
Monika Bhattacharjee, Arup Bose
The Annals of Statistics · 2016 · Volume 44 · Issue 2
Abstract
The existence of limiting spectral distribution (LSD) of $\hat{\Gamma}_{u}+\hat{\Gamma}_{u}^{*}$, the symmetric sum of the sample autocovariance matrix $\hat{\Gamma}_{u}$ 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 $\hat{\Gamma}_{u}+\hat{\Gamma}_{u}^{*}$, $\hat{\Gamma}_{u}\hat{\Gamma}_{u}^{*}$, $\hat{\Gamma}_{u}\hat{\Gamma}_{u}^{*}+\hat{\Gamma}_{k}\hat{\Gamma}_{k}^{*}$ 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.
Keywords
$*$-algebra, 37M10, 46L54, 58C40, 62J20, 62M10, asymptotically free, compound free Poisson, free cumulants, ID matrix, Infinite dimensional vector linear process, Limiting spectral distribution, Moment method, non-commutative probability space, non-crossing partitions, semi-circle law, Stieltjes transformation, symmetrized autocovariance matrices, Wigner matrix
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