Central Limit Theorem for Signal - to - Interference Ratio of Reduced Rank Linear Receiver

Let sk = 1 √ N (v1k, . . . , vNk) T , 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 = (SkPkS ∗ k + σ 2 I) and Akm = [sk,Rksk, . . . ,R m−1 k sk]. Define βkm = pks ∗ kAkm(A ∗ km ×RkAkm) −1 A ∗ kmsk, referred to as the signalto-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].