We study the behavior of a generalized 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. In addition, for any measurable cost function which is nonzero only when the rare event occurs, the method provides an unbiased estimator of the expected cost. On the other hand, the distribution of these points depends on the (random) number of points in the sample, and these points are not independent. So if we select at random one of the returned points, its distribution differs in general from the exact conditional distribution given the rare event. But 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.
The empirical distribution of the set of all points returned over all \(n\) replicates also converges to the conditional distribution given the rare event. The method also provides consistent confidence intervals for conditional expectations given that the rare event occurs.