An asymptotic analysis of a class of discrete nonparametric priors

In this paper we analyze the asymptotic behaviour of a large class of nonparametric priors, namely Gibbs-type priors, which represent a natural generalization of the Dirichlet process. After determining their topological support, we specifically investigate consistency of such priors according to the "what if", or frequentist, approach, which postulates the existence of a "true" distribution $P_0$. We provide a full taxonomy of their limiting behaviours: consistency holds essentially always for discrete $P_0$, whereas inconsistency may occur for diffuse $P_0$. Such findings are further illustrated by means of three specific priors admitting closed form expressions and exhibiting a wide range of asymptotic behaviours. For both Gibbs-type priors and discrete nonparametric priors in general, the possible inconsistency should not be interpreted as evidence against their use "tout court". It rather represents an indication that they are designed for modeling discrete distributions, at which consistency holds true, and a neat evidence against their use in the case of diffuse $P_0$.