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A new approach to multi-antenna downlink beam- forming is proposed that provides an improved robustness against uncertainty in the downlink channel covariance matrices caused by errors between the actual and estimated channel values. The proposed method uses the knowledge of the statistical distribution of such a covariance uncertainty to minimize the total downlink transmit power under the constraint that the outage probability does not exceed a certain threshold value. Although our approach initially leads to a non-convex optimization problem, it can be reformulated in a convex form using the semidefinite relaxation technique. The resulting convex optimization problem can be solved efficiently using the well-established interior point methods. Computer simulations verify performance improvements of the proposed technique as compared to the robust transmit beamforming method based on the worst-case performance optimization with judicious selection of the upper bounds on channel covariance errors.