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Let $\mathbf{s}_k=\frac{1}{\sqrt{N}}(v_{1k},...,v_{Nk})^T,$ with$\{v_{ik},i,k=1,...\}$ independent and identically distributed complex randomvariables. Write $\mathbf{S}_k=(\mathbf{s}_1,...,\mathbf{s}_{k-1},\mathbf{s}_{k+1},... ,\mathbf{s}_K),$ $\mathbf{P}_k=\operatorname{diag}(p_1,...,p_{k-1},p_{k+1},...,p_K)$,$\mathbf{R}_k=(\mathbf{S}_k\mathbf{P}_k\mathbf{S}_k^*+\sigma ^2\mathbf{I})$ and$\mathbf{A}_{km}=[\mathbf{s}_k,\mathbf{R}_k\mathbf{s}_k,...,\mathbf{R}_k^{m-1}\mathbf{s}_k]$. Define$\beta_{km}=p_k\mathbf{s}_k^*\mathbf{A}_{km}(\mathbf {A}_{km}^*\times\mathbf{R}_k\mathbf{A}_{km})^{-1}\mathbf{A}_{km}^*\mathbf{s}_k$, referred to asthe signal-to-interference ratio (SIR) of user $k$ under the multistage Wiener(MSW) receiver in a wireless communication system. It is proved that the outputSIR under the MSW and the mutual information statistic under the matched filter(MF) are both asymptotic Gaussian when $N/K\to c>0$. Moreover, we provide acentral limit theorem for linear spectral statistics of eigenvalues andeigenvectors of sample covariance matrices, which is a supplement of Theorem 2in Bai, Miao and Pan [Ann. Probab. 35 (2007) 1532--1572]. And we also improveTheorem 1.1 in Bai and Silverstein [Ann. Probab. 32 (2004) 553--605].

Central limit theorem for signal-to-interference ratio of reduced rank linear receiver