  We consider evaluation of proper posterior distributions obtained from
improper prior distributions. Our context is estimating a bounded function
$\phi$ of a parameter when the loss is quadratic. If the posterior mean of
$\phi$ is admissible for all bounded $\phi$, 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 concerns the $p$-dimensional multivariate normal distribution with
mean vector $\theta$ when the prior distribution has the form $g(\|\theta\|^2)
d\theta$ on the parameter space $\mathbb{R}^p$. Conditions on $g$ for strong
admissibility of the posterior are provided.
