Open Access
April 2016 Large sample behaviour of high dimensional autocovariance matrices
Monika Bhattacharjee, Arup Bose
Ann. Statist. 44(2): 598-628 (April 2016). DOI: 10.1214/15-AOS1378


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.


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Monika Bhattacharjee. Arup Bose. "Large sample behaviour of high dimensional autocovariance matrices." Ann. Statist. 44 (2) 598 - 628, April 2016.


Received: 1 June 2015; Revised: 1 August 2015; Published: April 2016
First available in Project Euclid: 17 March 2016

zbMATH: 1343.62053
MathSciNet: MR3476611
Digital Object Identifier: 10.1214/15-AOS1378

Primary: 62M10
Secondary: 37M10 , 46L54 , 58C40 , 62J20

Keywords: $*$-algebra , 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

Rights: Copyright © 2016 Institute of Mathematical Statistics

Vol.44 • No. 2 • April 2016
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