CLT for linear spectral statistics of large dimensional sample covariance matrices with dependent data

Tingting Zou (), Shurong Zheng (), Zhidong Bai, Jianfeng Yao and Hongtu Zhu
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Tingting Zou: Northeast Normal University
Shurong Zheng: Northeast Normal University
Zhidong Bai: Northeast Normal University
Jianfeng Yao: The University of Hong Kong
Hongtu Zhu: University of North Carolina at Chapel Hill

Statistical Papers, 2022, vol. 63, issue 2, No 12, 605-664

Abstract: Abstract This paper investigates the central limit theorem for linear spectral statistics of high dimensional sample covariance matrices of the form $${\mathbf {B}}_n=n^{-1}\sum _{j=1}^{n}{\mathbf {Q}}{\mathbf {x}}_j{\mathbf {x}}_j^{*}{\mathbf {Q}}^{*}$$ B n = n - 1 ∑ j = 1 n Q x j x j ∗ Q ∗ under the assumption that $$p/n\rightarrow y>0$$ p / n → y > 0 , where $${\mathbf {Q}}$$ Q is a $$p\times k$$ p × k nonrandom matrix and $$\{{\mathbf {x}}_j\}_{j=1}^n$$ { x j } j = 1 n is a sequence of independent k-dimensional random vector with independent entries. A key novelty here is that the dimension $$k\ge p$$ k ≥ p can be arbitrary, possibly infinity. This new model of sample covariance matrix $${\mathbf {B}}_n$$ B n covers most of the known models as its special cases. For example, standard sample covariance matrices are obtained with $$k=p$$ k = p and $${\mathbf {Q}}={\mathbf {T}}_n^{1/2}$$ Q = T n 1 / 2 for some positive definite Hermitian matrix $${\mathbf {T}}_n$$ T n . Also with $$k=\infty $$ k = ∞ our model covers the case of repeated linear processes considered in recent high-dimensional time series literature. The CLT found in this paper substantially generalizes the seminal CLT in Bai and Silverstein (Ann Probab 32(1):553–605, 2004). Applications of this new CLT are proposed for testing the AR(1) or AR(2) structure for a causal process. Our proposed tests are then used to analyze a large fMRI data set.

Keywords: Sample covariance matrices; Linear spectral statistics; Central limit theorem; Repeated linear processes; High-dimensional dependent data; 15B52; 62E20 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s00362-021-01250-3

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