Proof of the Outage Probability Conjecture for MISO Channels

It is conjectured that the covariance matrices minimizing the outage probability under a power constraint for multiple-input multiple-output channels with Gaussian fading are diagonal with either zeros or constant values on the diagonal. In the multiple-input single-output (MISO) setting, this is equivalent to conjecture that the Gaussian quadratic forms having largest tail probability correspond to such diagonal matrices. This paper provides a proof of the conjecture in this MISO setting.