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

Let , with {vik, i, k=1, …} independent and identically distributed complex random variables. Write Sk=(s1, …, sk-1, sk+1, …, sK), Pk=diag(p1, …, pk-1, pk+1, …, pK), Rk=(SkPkSk*+s2I) and Akm=[sk, Rksk, …, Rkm-1sk]. Define ßkm=pksk*Akm(Akm*×RkAkm)-1Akm*sk, referred to as the signal-to-interference ratio (SIR) of user k under the multistage Wiener (MSW) receiver in a wireless communication system. It is proved that the output SIR under the MSW and the mutual information statistic under the matched filter (MF) are both asymptotic Gaussian when N/K¿c>0. Moreover, we provide a central limit theorem for linear spectral statistics of eigenvalues and eigenvectors of sample covariance matrices, which is a supplement of Theorem 2 in Bai, Miao and Pan [Ann. Probab. 35 (2007) 1532–1572]. And we also improve Theorem 1.1 in Bai and Silverstein [Ann. Probab. 32 (2004) 553–605].