  We consider the secure transmission in ergodic fast-Rayleigh fading
multiple-input single-output single-antennaeavesdropper (MISOSE) wiretap
channels. We assume that the statistics of both the legitimate and eavesdropper
channels is the only available channel state information at the transmitter
(CSIT). By introducing a new secrecy capacity upper bound, we prove that the
secrecy capacity is achieved by Gaussian input without prefixing. To attain
this, we form another MISOSE channel for upper-bounding, and tighten the bound
by finding the worst correlations between the legitimate and eavesdropper
channel coefficients. The resulting upper bound is tighter than the others in
the literature which are based on modifying the correlation between the noises
at the legitimate receiver and eavesdropper. Next, we fully characterize the
ergodic secrecy capacity by showing that the optimal channel input covariance
matrix is a scaled identity matrix, with the transmit power allocated uniformly
among the antennas. The key to solve such a complicated stochastic optimization
problem is by exploiting the completely monotone property of the ergodic
secrecy capacity to use the stochastic ordering theory. Finally, our simulation
results show that for the considered channel setting, the secrecy capacity is
bounded in both the high signal-to-noise ratio and large number of transmit
antenna regimes.
