Kernel based Dirichlet sequences

Abstract

Let X=(X1,X2,…) be a sequence of random variables with values in a standard space (S,B). Suppose X1∼νandP(Xn+1∈⋅∣X1,…,Xn)=θν(⋅)+ ∑i=1nK(Xi)(⋅) n+θa.s. where θ>0 is a constant, ν a probability measure on B, and K a random probability measure on B. Then, X is exchangeable whenever K is a regular conditional distribution for ν given any sub-σ-field of B. Under this assumption, X enjoys all the main properties of classical Dirichlet sequences, including Sethuraman’s representation, conjugacy property, and convergence in total variation of predictive distributions. If μ is the weak limit of the empirical measures, conditions for μ to be a.s. discrete, or a.s. non-atomic, or μ≪ν a.s., are provided. Two CLT’s are proved as well. The first deals with stable convergence while the second concerns total variation distance.