Linear spectral statistics of sequential sample covariance matrices

Abstract

Independent $p$-dimensional vectors with independent complex or real valued entries such that $\mathbb{E}[{\mathbf{x}}_i]=0$, $\mathrm{Var}({\mathbf{x}}_i)={\mathbf{I}}_p$, $i=1,\dots ,n$, let ${\mathbf{T}}_n$ be a $p\times p$ Hermitian nonnegative definite matrix and $f$ be a given function. We prove that an approriately standardized version of the stochastic process $(\mathrm{tr}(f({\mathbf{B}}_{n,t})){)}_{t\in [t_0,1]}$ corresponding to a linear spectral statistic of the sequential empirical covariance estimator $$({\mathbf{B}}_{n,t}{)}_{t\in [t_0,1]}=(\frac{1}{n}\sum _{i=1}^{\lfloor nt\rfloor }{\mathbf{T}}_n^{1/2}{\mathbf{x}}_i{\mathbf{x}}_i^{\star }{\mathbf{T}}_n^{1/2}{)}_{t\in [t_0,1]}$$ converges weakly to a non-standard Gaussian process for $n,p\to \infty$. As an application we use these results to develop a novel approach for monitoring the sphericity assumption in a high-dimensional framework, even if the dimension of the underlying data is larger than the sample size.

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