Dispersion of infinite constellations in fast fading channels

In this work we extend the setting of communication without power constraint, proposed by Poltyrev, to fast fading channels with channel state information (CSI) at the receiver. The optimal codewords density, or actually the optimal normalized log density (NLD), is considered. Poltyrev's capacity for this channel is the highest achievable NLD, at possibly large block length, that guarantees a vanishing error probability. For a given finite block length n and a fixed error probability e, there is a gap between the highest achievable NLD and Poltyrev's capacity. As in other channels, this gap asymptotically vanishes as the square root of the channel dispersion V over n, multiplied by the inverse Q-function of the allowed error probability. This dispersion, derived in the paper, equals the dispersion of the power constrained fast fading channel at the high SNR regime. Connections to the error exponent of the peak power constrained fading channel are also discussed.