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This study mainly concentrates on analytical characterisation of the detection delays and false alarms in an online sequential probability test-based fault detection (FD) scheme. The probability distributions of the fault detection delay (FDD) and the time between false alarms (TBFA) are investigated. Analytical expressions of the probability density functions of FDD and TBFA are obtained. The results in this work are important for evaluation of FD performances in fault tolerant control systems, and they can also find useful applications in industrial alarm management systems.