Linear spectral statistics of sequential sample covariance matrices

Independent $p$-dimensional vectors with independent complex or real valued entries such that $\mathbb{E}[{\mathbf{x}}_i]=\mathbf{0}$, ${\rm Var } ($\mathbf{x}$_i) = $\mathbf{I}$_p$,$i=1, \ldots,n$,let$$\mathbf{T}$_n$bea$p \times p$Hermitiannonnegativedefinitematrixand$f $ be a given function. We prove that an approriately standardized version of the stochastic process $(\mathrm{tr}(f({\mathbf{B}}_{n,t})){)}_{t\in [t_0,1]}$ corresponding to a linear spectral statistic of the sequential empirical covariance estimator $$({\mathbf{B}}_{n,t}{)}_{t\in [t_0,1]}=(\frac{1}{n}\sum _{i=1}^{\lfloor nt\rfloor }{\mathbf{T}}_n^{1\mathrm{/}2}{\mathbf{x}}_i{\mathbf{x}}_i^{\star }{\mathbf{T}}_n^{1\mathrm{/}2}{)}_{t\in [t_0,1]}$$ converges weakly to a non-standard Gaussian process for $n,p\to \mathrm{\infty }$. 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.