  We consider beam selection using a fixed beamforming network (FBN) at a base
station with $M$ array antennas. In our setting, a Butler matrix is deployed at
the RF stage to form $M$ beams, and then the best beam is selected for
transmission. We provide the proofs of the key properties of the noncentral
chi-square distribution and the following properties of the beam selection gain
verifying that beam selection is superior to antenna selection in Rician
channels with any $K$-factors. Furthermore, we find asymptotically tight
stochastic bounds of the beam selection gain, which yield approximate closed
form expressions of the expected selection gain and the ergodic capacity. Beam
selection has the order of growth of the ergodic capacity
$\mathnormal{\Theta}(\log(M))$ regardless of user location in contrast to
$\mathnormal{\Theta}(\log(\log(M)))$ for antenna selection.
