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

Abstract

Let $\mathbf{s}_{k}=\frac{1}{\sqrt{N}}(v_{1k},\ldots,v_{Nk})^{T}$, with {v_ik, i, k=1, …} independent and identically distributed complex random variables. Write S_k=(s₁, …, s_k−1, s_k+1, …, s_K), P_k=diag(p₁, …, p_k−1, p_k+1, …, p_K), R_k=(S_kP_kS_k^*+σ²I) and A_km=[s_k, R_ks_k, …, R_k^m−1s_k]. Define β_km=p_ks_k^*A_km(A_km^*×R_kA_km)⁻¹A_km^*s_k, 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].