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Distributed signal detection schemes have received significant attention recently, but usually under the assumption of stationary observations which are independent from sensor to sensor. Here, order statistics based constant false alarm rate (OS-CFAR) detection techniques are applied to a distributed detection system with nonstationary observations where the signal observations are assumed to be dependent from sensor to sensor. Cases are considered where weak narrowband random signals are observed in additive Gaussian noise-plus-clutter of unknown power. Necessary conditions are given which specify the best sensor thresholds for some n-sensor cases. The best schemes using nonrandomized fusion rules are found for some specific two-sensor cases. The best schemes map use either AND or OR fusion rules depending on the specific false alarm probability and the number of reference observations used in the OS-CFAR scheme. Distributed OS-CFAR and cell-averaging CFAR (CA-CFAR) schemes are compared in terms of their capability to maintain false alarm probability in nonhomogeneous backgrounds. At least for the specific cases we have studied, there are OS-CFAR schemes which generally outperform the CA-CFAR schemes in this regard.