We study asymptotic behavior of Monte Carlo method. Local consistency is one of an ideal property of Monte Carlo method. However, it may fail to hold local consistency for several reason. In fact, in practice, it is more important to study such a non-ideal behavior. We call local degeneracy for one of a non-ideal behavior of Monte Carlo methods. We show some equivalent conditions for local degeneracy. As an application we study a Gibbs sampler (data augmentation) for cumulative logit model with or without marginal augmentation. It is well known that natural Gibbs sampler does not work well for this model. In a sense of local consistency and degeneracy, marginal augmentation is shown to improve the asymptotic property. However, when the number of categories is large, both methods are not locally consistent.