When there is no feedback from the fusion center to local sensors, we present a distributed Kalman filtering fusion formula for linear dynamic systems with sensor noises cross-correlated, and prove that under a mild condition the fused state estimate is equivalent to the centralized Kalman filtering using all sensor measurements, therefore, it achieves the best performance. Then, for the same dynamic system, when there is feedback, a modified Kalman filtering fusion with feedback for distributed recursive state estimators is proposed, and prove that the fusion formula with feedback is, as the fusion without feedback, still exactly equivalent to the corresponding centralized Kalman filtering fusion formula; the various P matrices in the feedback Kalman filtering at both local filters and the fusion center are still the covariance matrices of tracking errors; the feedback does reduce the covariance of each local tracking error. (c) 2007 Elsevier Ltd. All rights reserved.