We elaborate on the Hit-and-Run sampler, a Monte Carlo approach that estimates the value of a high-dimensional integral with integrand h(x)f(x) by sampling from a time-reversible Markov chain over the support of the density f. The Markov chain transitions are defined by choosing a random direction and then moving to a new point x whose likelihood depends on f in that direction. The serially dependent observations of h(xi) are averaged to estimate the integral. The sampler applies directly to f being a nonnegative function with finite integral. We generalize the convergence results of Belisle et al. to unbounded regions and to unbounded integrands. Here convergence is of the point estimator to the value of the integral; this convergence is based on convergence in distribution of realizations to their limiting distribution f. An important application is determining properties of Bayesian posterior distributions. Here f is proportional to the posterior density and h is chosen to indicate the property being estimated. Typical properties include means, variances, correlations, probabilities of regions, and predictive densities.
