  This paper focuses on optimal beamforming to maximize the mean
signal-to-noise ratio (SNR) for a reconfigurable intelligent surface
(RIS)-aided MISO downlink system under correlated Rician fading. The
beamforming problem becomes non-convex because of the unit modulus constraint
of passive RIS elements. To tackle this, we propose a semidefinite
relaxation-based iterative algorithm for obtaining statistically optimal
transmit beamforming vector and RIS-phase shift matrix. Further, we analyze the
outage probability (OP) and ergodic capacity (EC) to measure the performance of
the proposed beamforming scheme. Just like the existing works, the OP and EC
evaluations rely on the numerical computation of the iterative algorithm, which
does not clearly reveal the functional dependence of system performance on key
parameters. Therefore, we derive closed-form expressions for the optimal
beamforming vector and phase shift matrix along with their OP performance for
special cases of the general setup. Our analysis reveals that the i.i.d. fading
is more beneficial than the correlated case in the presence of LoS components.
This fact is analytically established for the setting in which the LoS is
blocked. Furthermore, we demonstrate that the maximum mean SNR improves
linearly/quadratically with the number of RIS elements in the absence/presence
of LoS component under i.i.d. fading.
