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We investigate diagnosability of stochastic discrete-event systems. We define the notions of A- and AA-diagnosability for stochastic automata; these notions are weaker than the corresponding notion of diagnosability for logical automata introduced by Sampath et al. Through the construction of a stochastic diagnoser, we determine offline conditions necessary and sufficient to guarantee A-diagnosability and sufficient to guarantee AA-diagnosability. We also show how the stochastic diagnoser can be used for on-line diagnosis of failure events. We illustrate the results through two examples from HVAC systems.