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Monitoring its health by detecting its failed sensors is essential to the reliable functioning of any sensor network. This paper presents a distributed, online, sequential algorithm for detecting multiple faults in a sensor network. The algorithm works by detecting change points in the correlation statistics of neighboring sensors, requiring only neighbors to exchange information. The algorithm provides guarantees on detection delay and false alarm probability. This appears to be the first work to offer such guarantees for a multiple sensor network. Based on the performance guarantees, we compute a tradeoff between sensor node density, detection delay and energy consumption. We also address synchronization, finite storage and data quantization. We validate our approach with some example applications.