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We consider the spectrum of the Sample Covariance matrix $\mathbf{A}_N:=\frac{\mathbf{X}_N \mathbf{X}_N^*}{N}, $ where $\mathbf{X}_N$ is the $P\timesN$ matrix with i.i.d. half-heavy tailed entries and $\frac{P}{N}\to y>0$ (theentries of the matrix have variance, but do not have the fourth moment). Wederive the Central Limit Theorem for the Stieltjes transform of the matrix$\mathbf{A}_N$ and compute the covariance kernel. Apart from that, we derivethe Central Limit Theorem for the Stieltjes transform of overlapping SampleCovariance matrices.

Central Limit Theorem for traces of the resolvents of half-heavy tailed Sample Covariance matrices