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Optimal transmitter designs obeying the water-filling principle are well-documented, and widely applied, when the propagation channel is deterministically known and regularly updated at the transmitter. Because channel state information (CSI) may be costly or impossible to acquire in rapidly varying wireless environments, we develop in this paper statistical water-filling approaches for stationary random fading channels. These approaches require only knowledge of the channel correlations that do not necessitate frequent updates, and can be easily acquired. Applied to a multiple transmit-antenna paradigm, our optimal transmitter design turns out to be an eigen-beamformer with multiple beams pointing to orthogonal directions along the eigenvectors of the channel's correlation matrix, and with proper power loading across the beams. The optimality pertains to minimizing a tight bound on the symbol error rate. The resulting loaded eigen-beamforming outperforms not only the equal-power allocation across all antennas, but also the conventional beamformer that transmits the available power along the strongest direction. Coupled with orthogonal space-time block codes, two-dimensional (2-D) eigen-beamforming emerges as a more attractive choice than conventional one-dimensional (1-D) beamforming with uniformly better performance, without rate reduction, and without complexity increase.