  The classical condition on the existence of uniformly exponentially
consistent tests for testing the true density against the complement of its
arbitrary neighborhood has been widely adopted in study of asymptotics of
Bayesian nonparametric procedures. Because we follow a Bayesian approach, it
seems to be more natural to explore alternative and appropriate conditions
which incorporate the prior distribution. In this paper we supply a new
prior-dependent integration condition to establish general posterior
convergence rate theorems for observations which may not be independent and
identically distributed. The posterior convergence rates for such observations
have recently studied by Ghosal and van der Vaart \cite{ghv1}. We moreover
adopt the Hausdorff $\alpha$-entropy given by Xing and Ranneby
\cite{xir1}\cite{xi1}, which is also prior-dependent and smaller than the
widely used metric entropies. These lead to extensions of several existing
theorems. In particular, we establish a posterior convergence rate theorem for
general Markov processes and as its application we improve on the currently
known posterior rate of convergence for a nonlinear autoregressive model.
