  We consider a fading channel in which a multi-antenna transmitter
communicates with a multi-antenna receiver through a reconfigurable intelligent
surface (RIS) that is made of $N$ reconfigurable passive scatterers impaired by
phase noise. The beamforming vector at the transmitter, the combining vector at
the receiver, and the phase shifts of the $N$ scatterers are optimized in order
to maximize the signal-to-noise-ratio (SNR) at the receiver. By assuming
Rayleigh fading (or line-of-sight propagation) on the transmitter-RIS link and
Rayleigh fading on the RIS-receiver link, we prove that the SNR is a random
variable that is equivalent in distribution to the product of three (or two)
independent random variables whose distributions are approximated by two (or
one) gamma random variables and the sum of two scaled non-central chi-square
random variables. The proposed analytical framework allows us to quantify the
robustness of RIS-aided transmission to fading channels. For example, we prove
that the amount of fading experienced on the transmitter-RIS-receiver channel
linearly decreases with $N$. This proves that RISs of large size can be
effectively employed to make fading less severe and wireless channels more
reliable.
