On numerical problems in simulations of highly reliable Markovian systems

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Simulation of highly reliable Markovian systems has been the subject of an extensive literature in recent years. Among all methods, simulation using importance sampling schemes gives the best results when the state space is large. In this paper, we highlight numerical problems that arise in rare events simulation, even when using importance sampling. The literature has up to now focused on variance reduction techniques, without any relation to the variance estimation for instance. The main contribution here is to relate the estimation of the considered parameter and of its variance to the bounded relative error and bounded normal approximation properties. We especially show that Bounded Normal Approximation implies that the variance is well-estimated, which implies Bounded Relative Error, implying itself that the parameter is well-estimated, but that no converse implication is true. This emphasizes the importance of Bounded Normal Approximation property, not frequently used in the literature yet.