On A Generalized Splitting Method For Sampling From A Conditional Distribution

We study the behavior of a splitting method for sampling from a given distribution conditional on the occurrence of a rare event. The method returns a random-sized sample of points such that unconditionally on the sample size, each point is distributed exactly according to the original distribution conditional on the rare event. For a cost function which is nonzero only when the rare event occurs, the method provides an unbiased estimator of the expected cost, but if we select at random one of the returned points, its distribution differs in general from the exact conditional distribution given the rare event. However, we prove that if we repeat the algorithm n times and select one of the returned points at random, the distribution of the selected point converges to the exact one in total variation when n increases.