Convergence of the conditional per symbol entropy for stationary Gaussian fading channels for almost all input sequences

In the present work, a discrete-time stationary Rayleigh flat-fading channel with unknown channel state information at the transmitter and the receiver is considered. The law of the channel is assumed to be known at the receiver and the fading process is supposed to be a stationary Gaussian process with an absolutely summable autocorrelation function. The conditional per symbol entropy of the output given the input is shown to converge to a constant for almost every realization of i.i.d. input variables. This implies the existence of the corresponding conditional entropy rate. Moreover, a novel inequality yielding a lower bound for the rate is derived.