On strong Hellinger consistency of posterior distributions

We establish a sufficient condition ensuring strong Hellinger consistency of posterior distributions. We also prove a strong Hellinger consistency theorem for the pseudoposterior distributions based on the likelihood ratio with power 0<α<1, which are introduced by Walker and Hjort [2001 ‘On Bayesian Consistency’, J. R. Statist. Soc., B 63, 811–821]. Our result is an extension of their theorem for α=½.