  In Bayesian decision theory, it is known that robustness with respect to the
loss and the prior can be improved by adding new observations. In this article
we study the rate of robustness improvement with respect to the number of
observations n. Three usual measures of posterior global robustness are
considered: the (range of the) Bayes actions set derived from a class of loss
functions, the maximum regret of using a particular loss when the subjective
loss belongs to a given class and the range of the posterior expected loss when
the loss function ranges over a class. We show that the rate of convergence of
the first measure of robustness is \sqrtn, while it is n for the other measures
under reasonable assumptions on the class of loss functions. We begin with the
study of two particular cases to illustrate our results.
