Monte Carlo methods for improper target distributions

Monte Carlo methods (based on iid sampling or Markov chains) for estimating integrals with respect to a proper target distribution (that is, a probability distribution) are well known in the statistics literature. If the target distribution π happens to be improper then it is shown here that the standard time average estimator based on Markov chains with π as its stationary distribution will converge to zero with probability 1, and hence is not appropriate. In this paper, we present some limit theorems for regenerative sequences and use these to develop some algorithms to produce strongly consistent estimators (called regeneration and ratio estimators) that work whether π is proper or improper. These methods may be referred to as regenerative sequence Monte Carlo (RSMC) methods. The regenerative sequences include Markov chains as a special case. We also present an algorithm that uses the domination of the given target π by a probability distribution π₀. Examples are given to illustrate the use and limitations of our algorithms.