  We study the asymptotic behavior of posterior distributions. We present
general posterior convergence rate theorems, which extend several results on
posterior convergence rates provided by Ghosal and Van der Vaart (2000), Shen
and Wasserman (2001) and Walker, Lijor and Prunster (2007). Our main tools are
the Hausdorff $\alpha$-entropy introduced by Xing and Ranneby (2008) and a new
notion of prior concentration, which is a slight improvement of the usual prior
concentration provided by Ghosal and Van der Vaart (2000). We apply our results
to several statistical models.
