Asymptotics of Detector Estimates
We next examine the consistency and asymptotic variance of the
estimates of the two subspace linear detectors. Assuming that the received
signal samples are independent and identically distributed (i.i.d.), then by the
strong law of large numbers, the sample mean 
converges
to Cr almost surely (a.s.) as the number of received
signals M 
. It then follows [521] that as 
a.s., for
k = 1, ..., K.
Therefore, we have
Equation 2.90
Equation 2.91
Similarly, 
a.s. as M . Hence both the estimated subspace linear
multiuser detectors based on the received signals are strongly consistent. However, it is in general biased
for finite number of samples. We next consider an asymptotic bound on the
estimation errors.
First, for all eigenvalues and the K largest eigenvectors of , the
following bounds hold a.s. [521, 609]:
Equation 2.92
Equation 2.93
Using the bounds above, we have
Equation 2.94
Note that 
, 
, and 
are all bounded. On the other hand, it is easily seen that
Equation 2.95
Equation 2.96
Therefore, we obtain the asymptotic estimation error for the
linear MMSE detector, and similarly that for the decorrelating detector, given,
respectively, by
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