Amenability: A survey for statistical applications of hunt-stein and related conditions on groups

Abstract

A number of conditions on groups have appeared in the literature of invariant statistical models in connection with minimaxity, approximation of invariant Bayes priors by proper priors, the relationship between Bayesian and classical inference, ergodic theorems, and other matters. In the last decade, rapid development has occurred in the field and many of these conditions are now known to be equivalent. We survey the subject, make the equivalences explicit, and list some groups of statistical interest which do, and also some which do not, have these properties. In particular, it is shown that the existence of the asymptotically invariant sequence of probabilities in the hypothesis of the Hunt-Stein theorem is equivalent to amenability, a condition that has been much studied by functional analysts. © 1981 Springer-Verlag.