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 $\mathbf{T }_n$ 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 $\big ( {\operatorname{tr}} ( f(\mathbf{B}_{n,t}) ) \big )_{t \in [t_0, 1]}$ corresponding to a linear spectral statistic of the sequential empirical covariance estimator $$ \big ( \mathbf{B}_{n,t} )_{t\in [ t_0 , 1]} = \Big ( \frac{1}{n} \sum_{i=1}^{\lfloor n t \rfloor} \mathbf{T }^{1/2}_n \mathbf{x}_i \mathbf{x}_i ^\star \mathbf{T }^{1/2}_n \Big)_{t\in [ t_0 , 1]} $$ 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.