Let sk=N1(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=(SkPkSk∗+σ2I) and
Akm=[sk,Rksk,...,Rkm−1sk]. Define
$\beta_{km}=p_k\mathbf{s}_k^*\mathbf{A}_{km}(\mathbf {A}_{km}^*\times\
mathbf{R}_k\mathbf{A}_{km})^{-1}\mathbf{A}_{km}^*\mathbf{s}_k,referredtoasthesignal−to−interferenceratio(SIR)ofuserkunderthemultistageWiener(MSW)receiverinawirelesscommunicationsystem.ItisprovedthattheoutputSIRundertheMSWandthemutualinformationstatisticunderthematchedfilter(MF)arebothasymptoticGaussianwhenN/K\to c>0$. 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].Comment: Published in at http://dx.doi.org/10.1214/07-AAP477 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Let Mn be an n×n real (resp. complex) Wigner matrix and
UnΛnUn∗ be its spectral decomposition. Set
(y1,y2...,yn)T=Un∗x, where x=(x1,x2,...,xn)T is a real (resp.
complex) unit vector. Under the assumption that the elements of Mn have 4
matching moments with those of GOE (resp. GUE), we show that the process
Xn(t)=2βn∑i=1⌊nt⌋(∣yi∣2−n1) converges weakly to the Brownian bridge for any
x such that ∣∣x∣∣∞→0 as n→∞,
where β=1 for the real case and β=2 for the complex case. Such a
result indicates that the othorgonal (resp. unitary) matrices with columns
being the eigenvectors of Wigner matrices are asymptotically Haar distributed
on the orthorgonal (resp. unitary) group from a certain perspective.Comment: typos correcte