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Under the high-dimensional setting that data dimension and sample size tendto infinity proportionally, we derive the central limit theorem (CLT) forlinear spectral statistics (LSS) of large-dimensional sample covariance matrix.Different from existing literature, our results do not require the assumptionthat the population covariance matrices are bounded. Moreover, many commonkernel functions in the real data such as logarithmic functions and polynomialfunctions are allowed in this paper. In our model, the number of spikedeigenvalues can be fixed or tend to infinity. One salient feature of theasymptotic mean and covariance in our proposed central limit theorem is that itis related to the divergence order of the population spectral norm.

CLT for LSS of sample covariance matrices with unbounded dispersions