We study asymptotic behavior of Markov chain Monte Carlo (MCMC) procedures. Sometimes the performances of MCMC procedures are poor and there are great importance for the study of such behavior. In this paper we call degeneracy for a particular type of poor performances. We show some equivalent conditions for degeneracy. As an application, we consider the cumulative probit model. It is well known that the natural data augmentation (DA) procedure does not work well for this model and the so-called...