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

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.