  The capacity of non-coherent stationary Gaussian fading channels with memory
under a peak-power constraint is studied in the asymptotic weak-signal regime.
It is assumed that the fading law is known to both transmitter and receiver but
that neither is cognizant of the fading realization. A connection is
demonstrated between the asymptotic behavior of channel capacity in this regime
and the asymptotic behavior of the prediction error incurred in predicting the
fading process from very noisy observations of its past. This connection can be
viewed as the low signal-to-noise ratio (SNR) analog of recent results by
Lapidoth & Moser and by Lapidoth demonstrating connections between the high SNR
capacity growth and the noiseless or almost-noiseless prediction error. We
distinguish between two families of fading laws: the ``slowly forgetting'' and
the ``quickly forgetting''. For channels in the former category the low SNR
capacity is achieved by IID inputs, whereas in the latter such inputs are
typically sub-optimal. Instead, the asymptotic capacity can be approached by
inputs with IID phase but block-constant magnitude.
