Linear spectral statistics of sequential sample covariance matrices
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Independent p-dimensional vectors with independent complex or real valued entries such that 𝔼 [𝐱_i] = 0, Var (𝐱_i) = 𝐈_p, i=1, …,n, let 𝐓 _n be a p × p Hermitian nonnegative definite matrix and f be a given function. We prove that an approriately standardized version of the stochastic process ( tr ( f(𝐁_n,t) ) )_t ∈ [t_0, 1] corresponding to a linear spectral statistic of the sequential empirical covariance estimator ( 𝐁_n,t )_t∈ [ t_0 , 1] = ( 1/n∑_i=1^⌊ n t ⌋𝐓 ^1/2_n 𝐱_i 𝐱_i ^⋆𝐓 ^1/2_n )_t∈ [ t_0 , 1] converges weakly to a non-standard Gaussian process for n,p→∞. 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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