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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

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.

Citation

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

Information

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

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

Rights: Copyright © 2016 Institute of Mathematical Statistics

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Vol.44 • No. 2 • April 2016
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