  We consider importance sampling as well as other properly weighted samples
with respect to a target distribution $\pi$ from a different point of view. By
considering the associated weights as sojourn times until the next jump, we
define appropriate jump processes. When the original sample sequence forms an
ergodic Markov chain, the associated jump process is an ergodic semi--Markov
process with stationary distribution $\pi$. Hence, the type of convergence of
properly weighted samples may be stronger than that of weighted means. In
particular, when the samples are independent and the mean weight is bounded
above, we describe a slight modification in order to achieve exact (weighted)
samples from the target distribution.
