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December 2019 Joint convergence of sample autocovariance matrices when $p/n\to 0$ with application
Monika Bhattacharjee, Arup Bose
Ann. Statist. 47(6): 3470-3503 (December 2019). DOI: 10.1214/18-AOS1785


Consider a high-dimensional linear time series model where the dimension $p$ and the sample size $n$ grow in such a way that $p/n\to 0$. Let $\hat{\Gamma }_{u}$ be the $u$th order sample autocovariance matrix. We first show that the LSD of any symmetric polynomial in $\{\hat{\Gamma }_{u},\hat{\Gamma }_{u}^{*},u\geq 0\}$ 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 $\{\hat{\Gamma }_{u},\hat{\Gamma }_{u}^{*},u\geq 0\}$. 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 high-dimensional MA process and in graphical and significance tests for hypotheses on coefficient matrices of one or several such independent processes.


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Monika Bhattacharjee. Arup Bose. "Joint convergence of sample autocovariance matrices when $p/n\to 0$ with application." Ann. Statist. 47 (6) 3470 - 3503, December 2019.


Received: 1 October 2017; Revised: 1 July 2018; Published: December 2019
First available in Project Euclid: 31 October 2019

Digital Object Identifier: 10.1214/18-AOS1785

Primary: 62M10
Secondary: 37M10 , 58C40

Keywords: asymptotic normality , estimation , Limiting spectral distribution , Moving average process , sample autocovariance matrices , testing of hypothesis , Trace

Rights: Copyright © 2019 Institute of Mathematical Statistics


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Vol.47 • No. 6 • December 2019
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