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In this paper, we consider a wireless communication scenario in which the channel output is marginally Gaussian, but not jointly Gaussian. In particular, we study the joint probability distribution of channel outputs in correlated Rayleigh fading channels in response to constant power signaling, such as M-ary phase shift keying (MPSK). We show that the distribution of the channel output at any given sampling time is marginally Gaussian. However, the joint distribution of a sequence of channel outputs cannot be jointly Gaussian. A consequence of this result is that the information rates stated to be exact in two recent contributions, are strict upper bounds to the achievable data rates. We examine the tightness of these upper bounds by comparing them with the MPSK upper bound under perfect channel state information (CSI) assumption. We find that the CSI upper bound is considerably tighter in slow fading channels, high signal-to-noise ratios, and low-dimension (such as binary) PSK signaling.