Evaluation of formal posterior distributions via Markov chain arguments

We consider evaluation of proper posterior distributions obtained from improper prior distributions. Our context is estimating a bounded function φ of a parameter when the loss is quadratic. If the posterior mean of φ is admissible for all bounded φ, the posterior is strongly admissible. We give sufficient conditions for strong admissibility. These conditions involve the recurrence of a Markov chain associated with the estimation problem. We develop general sufficient conditions for recurrence of general state space Markov chains that are also of independent interest. Our main example con-cerns the p-dimensional multivariate normal distribution with mean vector θ when the prior distribution has the form g(‖θ‖2) dθ on the parameter space R p. Conditions on g for strong admissibility of the posterior are provided